Question

which equation justifies why 10^(1/3) = ∛10? a) (10^(1/3))^3 = 10^(3*(1/3)) = 10 b) (10^(1/3))^3 = 10^(3-(1/3)) = 10 c) (10^(1/3))^3 = 10^(3÷(1/3)) = 10 d) (10^(1/3))^3 = 10^(3+(1/3)) = 10

Explanation:

Step1: Recall power - of - a - power rule

$(a^m)^n=a^{m\cdot n}$. For the expression $(10^{\frac{1}{3}})^3$, where $a = 10$, $m=\frac{1}{3}$, and $n = 3$.

Step2: Apply the rule

$(10^{\frac{1}{3}})^3=10^{\frac{1}{3}\cdot3}$.

Step3: Calculate the exponent

$\frac{1}{3}\cdot3 = 1$, so $10^{\frac{1}{3}\cdot3}=10^1=10$.

Answer:

The correct equation is the one that uses the power - of - a - power rule $(a^m)^n=a^{m\cdot n}$, so the correct option is the one where $(10^{\frac{1}{3}})^3=10^{(\frac{1}{3}\cdot3)} = 10$. Without seeing the full options clearly labeled as a, b, c, d in text, but based on the correct application of the rule, the correct simplification is based on the multiplication of exponents in the power - of - a - power rule. If we assume the options are based on different operations with the exponents, the correct one is the one with the multiplication of $\frac{1}{3}$ and 3 in the exponent for the transformation from $(10^{\frac{1}{3}})^3$ to $10^{(\frac{1}{3}\cdot3)}$.