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

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

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Simplify the product of two radical expressions having same variable

Full solution

Q. Which expressions are equivalent to

(\sqrt[4]{d})^{3}

?

\newline

Choose all answers that apply:

\newline

A

\sqrt[4]{d^{3}}

\newline

B

\left(d^{\frac{1}{4}}\right)^{3}

\newline

c

\sqrt[3]{d^{4}}

\newline

D None of the above

Express in Exponential Form: First, let's express the given expression $(\sqrt[4]{d})^{3}$ in exponential form using the property that $\sqrt[n]{a} = a^{\frac{1}{n}}$ . $\newline$ $(\sqrt[4]{d})^{3} = (d^{\frac{1}{4}})^{3}$
Apply Power Rule: Now, apply the power rule for exponents, which states that $(a^{m})^{n} = a^{m*n}$ , to simplify the expression. $(d^{\frac{1}{4}})^{3} = d^{(\frac{1}{4}*3)} = d^{\frac{3}{4}}$
Check Answer Choices: Let's check each answer choice to see if it is equivalent to $d^{(3/4)}$ . $\newline$ For choice A: $\sqrt[4]{d^{3}} = d^{(3/4)}$ because the fourth root of $d$ cubed is the same as raising $d$ to the power of $3/4$ .
Choice A Verification: For choice B: $(d^{(1/4)})^{3} = d^{3/4}$ by the power rule for exponents, as we have already calculated.
Choice B Verification: For choice C: $\sqrt[3]{d^{4}}$ is not equivalent because it would be expressed as $d^{\frac{4}{3}}$ in exponential form, which is different from $d^{\frac{3}{4}}$ .
Choice C Verification: For choice D: ＂None of the above＂ is not correct because we have found that choices A and B are equivalent to the given expression.

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