Asked by n
Two equations are shown.
Equation 1: a2
= 64
Equation 2: b3
= 64
Explain why the value of a is different than the value of b using written sentences and math calculations.
All Answers 1
Answered by
GPT-5 mini
AI
Equation 1: a^2 = 64
Equation 2: b^3 = 64
Solve 1: a^2 = 64
Take square roots: |a| = 8, so a = 8 or a = −8. (Check: 8^2 = 64 and (−8)^2 = 64.)
Solve 2: b^3 = 64
Take cube root: b = ³√64 = 4. (Check: 4^3 = 64; (−4)^3 = −64, so −4 is not a solution.)
Why they differ (in words): raising to an even power (2) loses the sign information, so both positive and negative numbers can give the same result (hence two real solutions ±8). Raising to an odd power (3) preserves sign, so there is exactly one real number whose cube is 64 (namely 4).
Equation 2: b^3 = 64
Solve 1: a^2 = 64
Take square roots: |a| = 8, so a = 8 or a = −8. (Check: 8^2 = 64 and (−8)^2 = 64.)
Solve 2: b^3 = 64
Take cube root: b = ³√64 = 4. (Check: 4^3 = 64; (−4)^3 = −64, so −4 is not a solution.)
Why they differ (in words): raising to an even power (2) loses the sign information, so both positive and negative numbers can give the same result (hence two real solutions ±8). Raising to an odd power (3) preserves sign, so there is exactly one real number whose cube is 64 (namely 4).
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