Which expressions are equivalent to (root(4)(d))^(3) ? Choose all answers that apply: A root(4)(d^(3)) B root(3)(d^(4)) C (d^((1)/(4)))^(3) D None of the above

Question

Which expressions are equivalent to  (root(4)(d))^(3) ? Choose all answers that apply: A  root(4)(d^(3)) B  root(3)(d^(4)) C  (d^((1)/(4)))^(3) D None of the above

Bytelearn · Accepted Answer

Express in Exponential Form: First, let's express the given expression (root(4)(d))^(3) in exponential form using the property that root(n)(a) = a^(1/n). (root(4)(d))^(3) = (d^(1/4))^3 Now, we apply the power to a power rule, which states that (a^(m))^n = a^(m*n). (d^(1/4))^3 = d^((1/4)*3) = d^(3/4)Consider Option A: Now let's consider option A: root(4)(d^(3)). We convert this to exponential form: root(4)(d^(3)) = (d^(3))^(1/4). Applying the power to a power rule: (d^(3))^(1/4) = d^(3/4). This is the same as our expression from step 1, so option A is equivalent.Consider Option B: Next, let's consider option B: root(3)(d^(4)). We convert this to exponential form: root(3)(d^(4)) = (d^(4))^(1/3). Applying the power to a power rule: (d^(4))^(1/3) = d^(4/3). This is not the same as our expression from step 1, so option B is not equivalent.Consider Option C: Now let's consider option C: (d^((1)/(4)))^(3). This is the same as our initial expression from step 1, so option C is equivalent.Option D: Option D states ＂None of the above,＂ which is not correct because we have found that options A and C are equivalent to the given expression.

Which expressions are equivalent to $(\sqrt[4]{d})^{3}$ ? $\newline$ Choose all answers that apply: $\newline$ (A) $\sqrt[4]{d^{3}}$ $\newline$ (B) $\sqrt[3]{d^{4}}$ $\newline$ (C) $\left(d^{\frac{1}{4}}\right)^{3}$ $\newline$ (D) None of the above

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Which expressions are equivalent to $(\sqrt[4]{d})^{3}$ ? $\newline$ Choose all answers that apply: $\newline$ (A) $\sqrt[4]{d^{3}}$ $\newline$ (B) $\sqrt[3]{d^{4}}$ $\newline$ (C) $\left(d^{\frac{1}{4}}\right)^{3}$ $\newline$ (D) None of the above