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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261003-r01-power-root-foundations · 2026-10-03 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (-12)^0 exactly.
1
(-12)^0=1. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-exact · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (-11)^1 exactly.
-11
(-11)^1=-11. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-exact · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (-10)^2 exactly.
100
(-10)^2=100. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-exact · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (-9)^3 exactly.
-729
(-9)^3=-729. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-exact · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (-8)^4 exactly.
4096
(-8)^4=4096. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-exact · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (-7)^5 exactly.
-16807
(-7)^5=-16807. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-exact · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (-6)^0 exactly.
1
(-6)^0=1. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-exact · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (-5)^1 exactly.
-5
(-5)^1=-5. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-exact · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (-4)^2 exactly.
16
(-4)^2=16. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-exact · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (-3)^3 exactly.
-27
(-3)^3=-27. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-exact · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (-2)^4 exactly.
16
(-2)^4=16. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-exact · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (-1)^5 exactly.
-1
(-1)^5=-1. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-exact · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (0)^0 exactly.
1
(0)^0=1. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-exact · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (0)^1 exactly.
0
(0)^1=0. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-exact · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (2)^2 exactly.
4
(2)^2=4. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-exact · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (3)^3 exactly.
27
(3)^3=27. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-exact · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (4)^4 exactly.
256
(4)^4=256. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-exact · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (5)^5 exactly.
3125
(5)^5=3125. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-exact · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (6)^0 exactly.
1
(6)^0=1. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-exact · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (7)^1 exactly.
7
(7)^1=7. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-exact · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (8)^2 exactly.
64
(8)^2=64. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-exact · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (9)^3 exactly.
729
(9)^3=729. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-exact · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (10)^4 exactly.
10000
(10)^4=10000. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-exact · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (11)^5 exactly.
161051
(11)^5=161051. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-exact · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Compute (12)^0 exactly.
1
(12)^0=1. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-exact · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (0)^0 positive, negative, or zero?
positive
(0)^0=1. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-sign · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (-9)^1 positive, negative, or zero?
negative
(-9)^1=-9. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-sign · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (-8)^2 positive, negative, or zero?
positive
(-8)^2=64. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-sign · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (-7)^3 positive, negative, or zero?
negative
(-7)^3=-343. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-sign · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (-6)^4 positive, negative, or zero?
positive
(-6)^4=1296. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-sign · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (-5)^5 positive, negative, or zero?
negative
(-5)^5=-3125. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-sign · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (-4)^6 positive, negative, or zero?
positive
(-4)^6=4096. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-sign · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (-3)^0 positive, negative, or zero?
positive
(-3)^0=1. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-sign · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (-2)^1 positive, negative, or zero?
negative
(-2)^1=-2. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-sign · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (-1)^2 positive, negative, or zero?
positive
(-1)^2=1. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-sign · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (0)^3 positive, negative, or zero?
zero
(0)^3=0. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-sign · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (1)^4 positive, negative, or zero?
positive
(1)^4=1. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-sign · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (2)^5 positive, negative, or zero?
positive
(2)^5=32. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-sign · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (3)^6 positive, negative, or zero?
positive
(3)^6=729. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-sign · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (4)^0 positive, negative, or zero?
positive
(4)^0=1. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-sign · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (5)^1 positive, negative, or zero?
positive
(5)^1=5. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-sign · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (6)^2 positive, negative, or zero?
positive
(6)^2=36. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-sign · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (7)^3 positive, negative, or zero?
positive
(7)^3=343. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-sign · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (8)^4 positive, negative, or zero?
positive
(8)^4=4096. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-sign · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (9)^5 positive, negative, or zero?
positive
(9)^5=59049. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-sign · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (10)^6 positive, negative, or zero?
positive
(10)^6=1000000. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-sign · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (11)^0 positive, negative, or zero?
positive
(11)^0=1. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-sign · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (12)^1 positive, negative, or zero?
positive
(12)^1=12. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-sign · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (13)^2 positive, negative, or zero?
positive
(13)^2=169. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-sign · candidate
For real bases and natural-number exponents, use the empty-product convention a^0=1, including 0^0=1 in this algebraic setting. Is (14)^3 positive, negative, or zero?
positive
(14)^3=2744. At exponent zero the empty product is 1; for positive exponents use the base sign and exponent parity. This convention is not a rule for limits.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-natural-power-sign · candidate
Using the principal square root on the nonnegative reals, find the exact nonnegative value of √0.
0
0²=0 and 0≥0, so the principal square root is 0; it is a single value rather than two signed branches.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-exact · candidate
Using the principal square root on the nonnegative reals, find the exact nonnegative value of √1.
1
1²=1 and 1≥0, so the principal square root is 1; it is a single value rather than two signed branches.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-exact · candidate
Using the principal square root on the nonnegative reals, find the exact nonnegative value of √4.
2
2²=4 and 2≥0, so the principal square root is 2; it is a single value rather than two signed branches.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-exact · candidate
Using the principal square root on the nonnegative reals, find the exact nonnegative value of √9.
3
3²=9 and 3≥0, so the principal square root is 3; it is a single value rather than two signed branches.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-exact · candidate
Using the principal square root on the nonnegative reals, find the exact nonnegative value of √16.
4
4²=16 and 4≥0, so the principal square root is 4; it is a single value rather than two signed branches.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-exact · candidate
Using the principal square root on the nonnegative reals, find the exact nonnegative value of √25.
5
5²=25 and 5≥0, so the principal square root is 5; it is a single value rather than two signed branches.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-exact · candidate
Using the principal square root on the nonnegative reals, find the exact nonnegative value of √36.
6
6²=36 and 6≥0, so the principal square root is 6; it is a single value rather than two signed branches.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-exact · candidate
Using the principal square root on the nonnegative reals, find the exact nonnegative value of √49.
7
7²=49 and 7≥0, so the principal square root is 7; it is a single value rather than two signed branches.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-exact · candidate
Using the principal square root on the nonnegative reals, find the exact nonnegative value of √64.
8
8²=64 and 8≥0, so the principal square root is 8; it is a single value rather than two signed branches.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-exact · candidate
Using the principal square root on the nonnegative reals, find the exact nonnegative value of √81.
9
9²=81 and 9≥0, so the principal square root is 9; it is a single value rather than two signed branches.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-exact · candidate
Using the principal square root on the nonnegative reals, find the exact nonnegative value of √100.
10
10²=100 and 10≥0, so the principal square root is 10; it is a single value rather than two signed branches.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-exact · candidate
Using the principal square root on the nonnegative reals, find the exact nonnegative value of √121.
11
11²=121 and 11≥0, so the principal square root is 11; it is a single value rather than two signed branches.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-exact · candidate
Using the principal square root on the nonnegative reals, find the exact nonnegative value of √144.
12
12²=144 and 12≥0, so the principal square root is 12; it is a single value rather than two signed branches.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-exact · candidate
Using the principal square root on the nonnegative reals, find the exact nonnegative value of √169.
13
13²=169 and 13≥0, so the principal square root is 13; it is a single value rather than two signed branches.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-exact · candidate
Using the principal square root on the nonnegative reals, find the exact nonnegative value of √196.
14
14²=196 and 14≥0, so the principal square root is 14; it is a single value rather than two signed branches.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-exact · candidate
Using the principal square root on the nonnegative reals, find the exact nonnegative value of √225.
15
15²=225 and 15≥0, so the principal square root is 15; it is a single value rather than two signed branches.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-exact · candidate
Using the principal square root on the nonnegative reals, find the exact nonnegative value of √256.
16
16²=256 and 16≥0, so the principal square root is 16; it is a single value rather than two signed branches.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-exact · candidate
Using the principal square root on the nonnegative reals, find the exact nonnegative value of √289.
17
17²=289 and 17≥0, so the principal square root is 17; it is a single value rather than two signed branches.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-exact · candidate
Using the principal square root on the nonnegative reals, find the exact nonnegative value of √324.
18
18²=324 and 18≥0, so the principal square root is 18; it is a single value rather than two signed branches.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-exact · candidate
Using the principal square root on the nonnegative reals, find the exact nonnegative value of √361.
19
19²=361 and 19≥0, so the principal square root is 19; it is a single value rather than two signed branches.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-exact · candidate
Using the principal square root on the nonnegative reals, find the exact nonnegative value of √400.
20
20²=400 and 20≥0, so the principal square root is 20; it is a single value rather than two signed branches.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-exact · candidate
Using the principal square root on the nonnegative reals, find the exact nonnegative value of √441.
21
21²=441 and 21≥0, so the principal square root is 21; it is a single value rather than two signed branches.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-exact · candidate
Using the principal square root on the nonnegative reals, find the exact nonnegative value of √484.
22
22²=484 and 22≥0, so the principal square root is 22; it is a single value rather than two signed branches.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-exact · candidate
Using the principal square root on the nonnegative reals, find the exact nonnegative value of √529.
23
23²=529 and 23≥0, so the principal square root is 23; it is a single value rather than two signed branches.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-exact · candidate
Using the principal square root on the nonnegative reals, find the exact nonnegative value of √576.
24
24²=576 and 24≥0, so the principal square root is 24; it is a single value rather than two signed branches.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-exact · candidate
The real principal square root in this exercise is defined only for nonnegative inputs. Does r=-12 belong to its domain? Answer Yes or No.
No
r=-12 does not satisfy r≥0, so the answer is No. A software convention returning zero for a negative input is outside this definition.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-domain · candidate
The real principal square root in this exercise is defined only for nonnegative inputs. Does r=-11 belong to its domain? Answer Yes or No.
No
r=-11 does not satisfy r≥0, so the answer is No. A software convention returning zero for a negative input is outside this definition.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-domain · candidate
The real principal square root in this exercise is defined only for nonnegative inputs. Does r=-10 belong to its domain? Answer Yes or No.
No
r=-10 does not satisfy r≥0, so the answer is No. A software convention returning zero for a negative input is outside this definition.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-domain · candidate
The real principal square root in this exercise is defined only for nonnegative inputs. Does r=-9 belong to its domain? Answer Yes or No.
No
r=-9 does not satisfy r≥0, so the answer is No. A software convention returning zero for a negative input is outside this definition.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-domain · candidate
The real principal square root in this exercise is defined only for nonnegative inputs. Does r=-8 belong to its domain? Answer Yes or No.
No
r=-8 does not satisfy r≥0, so the answer is No. A software convention returning zero for a negative input is outside this definition.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-domain · candidate
The real principal square root in this exercise is defined only for nonnegative inputs. Does r=-7 belong to its domain? Answer Yes or No.
No
r=-7 does not satisfy r≥0, so the answer is No. A software convention returning zero for a negative input is outside this definition.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-domain · candidate
The real principal square root in this exercise is defined only for nonnegative inputs. Does r=-6 belong to its domain? Answer Yes or No.
No
r=-6 does not satisfy r≥0, so the answer is No. A software convention returning zero for a negative input is outside this definition.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-domain · candidate
The real principal square root in this exercise is defined only for nonnegative inputs. Does r=-5 belong to its domain? Answer Yes or No.
No
r=-5 does not satisfy r≥0, so the answer is No. A software convention returning zero for a negative input is outside this definition.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-domain · candidate
The real principal square root in this exercise is defined only for nonnegative inputs. Does r=-4 belong to its domain? Answer Yes or No.
No
r=-4 does not satisfy r≥0, so the answer is No. A software convention returning zero for a negative input is outside this definition.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-domain · candidate
The real principal square root in this exercise is defined only for nonnegative inputs. Does r=-3 belong to its domain? Answer Yes or No.
No
r=-3 does not satisfy r≥0, so the answer is No. A software convention returning zero for a negative input is outside this definition.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-domain · candidate
The real principal square root in this exercise is defined only for nonnegative inputs. Does r=-2 belong to its domain? Answer Yes or No.
No
r=-2 does not satisfy r≥0, so the answer is No. A software convention returning zero for a negative input is outside this definition.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-domain · candidate
The real principal square root in this exercise is defined only for nonnegative inputs. Does r=-1 belong to its domain? Answer Yes or No.
No
r=-1 does not satisfy r≥0, so the answer is No. A software convention returning zero for a negative input is outside this definition.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-domain · candidate
The real principal square root in this exercise is defined only for nonnegative inputs. Does r=0 belong to its domain? Answer Yes or No.
Yes
r=0 satisfies r≥0, so the answer is Yes. A software convention returning zero for a negative input is outside this definition.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-domain · candidate
The real principal square root in this exercise is defined only for nonnegative inputs. Does r=1 belong to its domain? Answer Yes or No.
Yes
r=1 satisfies r≥0, so the answer is Yes. A software convention returning zero for a negative input is outside this definition.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-domain · candidate
The real principal square root in this exercise is defined only for nonnegative inputs. Does r=2 belong to its domain? Answer Yes or No.
Yes
r=2 satisfies r≥0, so the answer is Yes. A software convention returning zero for a negative input is outside this definition.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-domain · candidate
The real principal square root in this exercise is defined only for nonnegative inputs. Does r=3 belong to its domain? Answer Yes or No.
Yes
r=3 satisfies r≥0, so the answer is Yes. A software convention returning zero for a negative input is outside this definition.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-domain · candidate
The real principal square root in this exercise is defined only for nonnegative inputs. Does r=4 belong to its domain? Answer Yes or No.
Yes
r=4 satisfies r≥0, so the answer is Yes. A software convention returning zero for a negative input is outside this definition.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-domain · candidate
The real principal square root in this exercise is defined only for nonnegative inputs. Does r=5 belong to its domain? Answer Yes or No.
Yes
r=5 satisfies r≥0, so the answer is Yes. A software convention returning zero for a negative input is outside this definition.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-domain · candidate
The real principal square root in this exercise is defined only for nonnegative inputs. Does r=6 belong to its domain? Answer Yes or No.
Yes
r=6 satisfies r≥0, so the answer is Yes. A software convention returning zero for a negative input is outside this definition.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-domain · candidate
The real principal square root in this exercise is defined only for nonnegative inputs. Does r=7 belong to its domain? Answer Yes or No.
Yes
r=7 satisfies r≥0, so the answer is Yes. A software convention returning zero for a negative input is outside this definition.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-domain · candidate
The real principal square root in this exercise is defined only for nonnegative inputs. Does r=8 belong to its domain? Answer Yes or No.
Yes
r=8 satisfies r≥0, so the answer is Yes. A software convention returning zero for a negative input is outside this definition.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-domain · candidate
The real principal square root in this exercise is defined only for nonnegative inputs. Does r=9 belong to its domain? Answer Yes or No.
Yes
r=9 satisfies r≥0, so the answer is Yes. A software convention returning zero for a negative input is outside this definition.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-domain · candidate
The real principal square root in this exercise is defined only for nonnegative inputs. Does r=10 belong to its domain? Answer Yes or No.
Yes
r=10 satisfies r≥0, so the answer is Yes. A software convention returning zero for a negative input is outside this definition.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-domain · candidate
The real principal square root in this exercise is defined only for nonnegative inputs. Does r=11 belong to its domain? Answer Yes or No.
Yes
r=11 satisfies r≥0, so the answer is Yes. A software convention returning zero for a negative input is outside this definition.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-domain · candidate
The real principal square root in this exercise is defined only for nonnegative inputs. Does r=12 belong to its domain? Answer Yes or No.
Yes
r=12 satisfies r≥0, so the answer is Yes. A software convention returning zero for a negative input is outside this definition.
Reviewed step-by-step derivation
Natural-number powers and principal square roots · qf-principal-square-root-domain · candidate
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