Which expressions are equivalent to (d^((1)/(8)))^(5) ? Choose all answers that apply: (A) (d^(5))^((1)/(8)) (B) (d^(5))^(8) (C) (root(8)(d))^(5) (D) None of the above

Evaluate Power Rule: Evaluate the given expression using the power of a power rule. The power of a power rule states that (a^(m))^n = a^(m*n). So, (d^((1)/(8)))^(5) = d^((1)/(8)*5) = d^(5/8).Check Option A: Check option A: (d^(5))^((1)/(8)). Using the power of a power rule, (d^(5))^((1)/(8)) = d^(5*(1/8)) = d^(5/8). This is equivalent to the original expression.Check Option B: Check option B: (d^(5))^(8). Using the power of a power rule, (d^(5))^(8) = d^(5*8) = d^(40). This is not equivalent to the original expression.Check Option C: Check option C: (root(8)(d))^(5). The expression (root(8)(d))^(5) means taking the 8th root of d and then raising it to the 5th power. The 8th root of d is d^(1/8), so (root(8)(d))^(5) = (d^(1/8))^5. Using the power of a power rule, (d^(1/8))^5 = d^((1/8)*5) = d^(5/8). This is equivalent to the original expression.Determine Final Answer: Determine the final answer. The expressions equivalent to (d^((1)/(8)))^(5) are: A (d^(5))^((1)/(8)) and C (root(8)(d))^(5).

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Which expressions are equivalent to (d^((1)/(8)))^(5) ?
Choose all answers that apply:
(A) (d^(5))^((1)/(8))
(B) (d^(5))^(8)
(C) (root(8)(d))^(5)
(D) None of the above

Which expressions are equivalent to $(d^{\frac{1}{8}})^5$ ? $\newline$ Choose all answers that apply: $\newline$ $(A)$ $(d^5)^{\frac{1}{8}}$ $\newline$ $(B)$ $(d^5)^8$ $\newline$ $(C)$ $\sqrt[8]{d}^5$ $\newline$ $(D)$ None of the above

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Q. Which expressions are equivalent to

(d^{\frac{1}{8}})^5

\newline

Choose all answers that apply:

\newline

(A)

(d^5)^{\frac{1}{8}}

\newline

(B)

(d^5)^8

\newline

(C)

\sqrt[8]{d}^5

\newline

(D)

None of the above

Evaluate Power Rule: Evaluate the given expression using the power of a power rule. $\newline$ The power of a power rule states that $(a^{m})^{n} = a^{m*n}$ . $\newline$ So, $(d^{\frac{1}{8}})^{5} = d^{\frac{1}{8}*5} = d^{\frac{5}{8}}$ .
Check Option A: Check option A: $(d^{5})^{(1/8)}$ . Using the power of a power rule, $(d^{5})^{(1/8)} = d^{5*(1/8)} = d^{5/8}$ . This is equivalent to the original expression.
Check Option B: Check option B: $(d^{5})^{8}$ . Using the power of a power rule, $(d^{5})^{8} = d^{5\times8} = d^{40}$ . This is not equivalent to the original expression.
Check Option C: Check option C: $(\sqrt[8]{d})^{5}$ . The expression $(\sqrt[8]{d})^{5}$ means taking the $8$ th root of $d$ and then raising it to the $5$ th power. The $8$ th root of $d$ is $d^{1/8}$ , so $(\sqrt[8]{d})^{5} = (d^{1/8})^5$ . Using the power of a power rule, $(d^{1/8})^5 = d^{(1/8)*5} = d^{5/8}$ . This is equivalent to the original expression.
Determine Final Answer: Determine the final answer. $\newline$ The expressions equivalent to $(d^{(1/8)})^{5}$ are: $\newline$ A $(d^{5})^{(1/8)}$ and C $(\sqrt[8]{d})^{5}$ .