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

View step-by-step help

Home

Math Problems

Algebra 1

Identify equivalent linear expressions

Full solution

Q. Which expressions are equivalent to

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

\newline

Choose all answers that apply:

\newline

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

\newline

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

\newline

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

\newline

D None of the above

Understand Given Expression: First, let's understand the given expression $(\sqrt[4]{d})^{3}$ . This means we are taking the fourth root of $d$ and then raising it to the power of $3$ .
Express Fourth Root as Exponent: Now, let's express the fourth root of $d$ as an exponent. The fourth root of $d$ is the same as $d$ raised to the power of $\frac{1}{4}$ . $\newline$ $\sqrt[4]{d}^3 = (d^{\frac{1}{4}})^3$
Apply Power Rule for Exponents: Next, we apply the power rule for exponents, which states that $(a^{m})^{n} = a^{m*n}$ . So, we multiply the exponents $\frac{1}{4}$ and $3$ . $(d^{\frac{1}{4}})^{3} = d^{\frac{1}{4} * 3} = d^{\frac{3}{4}}$
Compare with Given Options: Now, let's compare the result with the given options: $\newline$ A. $\sqrt[4]{d^{3}}$ is not equivalent because it represents the fourth root of $d$ cubed, not $d$ to the power of $\frac{3}{4}$ .
Compare with Given Options: Now, let's compare the result with the given options: $\newline$ A. $\sqrt[4]{d^3}$ is not equivalent because it represents the fourth root of $d$ cubed, not $d$ to the power of $\frac{3}{4}$ .B. $(d^{\frac{1}{4}})^3$ is equivalent because it is the expression we derived, which is $d^{\frac{3}{4}}$ .
Compare with Given Options: Now, let's compare the result with the given options: $\newline$ A. $\sqrt[4]{d^{3}}$ is not equivalent because it represents the fourth root of $d$ cubed, not $d$ to the power of $\frac{3}{4}$ .B. $(d^{\frac{1}{4}})^3$ is equivalent because it is the expression we derived, which is $d^{\frac{3}{4}}$ .C. $\sqrt[3]{d^{4}}$ is not equivalent because it represents the cube root of $d$ to the fourth power, not $d$ to the power of $\frac{3}{4}$ .
Compare with Given Options: Now, let's compare the result with the given options: $\newline$ A. $\sqrt[4]{d^{3}}$ is not equivalent because it represents the fourth root of $d$ cubed, not $d$ to the power of $\frac{3}{4}$ .B. $(d^{\frac{1}{4}})^3$ is equivalent because it is the expression we derived, which is $d^{\frac{3}{4}}$ .C. $\sqrt[3]{d^{4}}$ is not equivalent because it represents the cube root of $d$ to the fourth power, not $d$ to the power of $\frac{3}{4}$ .D. ＂None of the above＂ is not correct because we have already found that option B is equivalent to the given expression.