write each expression with a single rational exponent. show each step of your process. which expressions are…

Explanation:

Step1: Rewrite A using exponent rules

Recall that $\sqrt[n]{a}=a^{\frac{1}{n}}$. So, $\sqrt[4]{x^{3}} = x^{\frac{3}{4}}$.

Step2: Rewrite B using negative - exponent rule

Recall that $a^{-n}=\frac{1}{a^{n}}$. So, $\frac{1}{x^{-1}}=x^{1}$.

Step3: Simplify the exponent in C

First, use the rule $a^{m}\cdot a^{n}=a^{m + n}$ for the terms inside the radical: $x^{5}\cdot x^{4}\cdot x^{2}=x^{5 + 4+2}=x^{11}$. Then, $\sqrt[10]{x^{11}}=x^{\frac{11}{10}}$.

Step4: Simplify the exponent in D

Use the rule $a^{m}\cdot a^{n}=a^{m + n}$: $x^{\frac{1}{3}}\cdot x^{\frac{1}{3}}\cdot x^{\frac{1}{3}}=x^{\frac{1 + 1+1}{3}}=x^{1}$.

To find equivalent expressions:

• Expressions B and D are equivalent because $\frac{1}{x^{-1}}=x^{1}$ and $x^{\frac{1}{3}}\cdot x^{\frac{1}{3}}\cdot x^{\frac{1}{3}}=x^{1}$.

Answer:

A. $x^{\frac{3}{4}}$ B. $x^{1}$ C. $x^{\frac{11}{10}}$ D. $x^{1}$ Equivalent expressions: B and D.