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Which expressions are equivalent to
x^((4)/(5)) ?
Choose all answers that apply:
A
(root(4)(x))^(5)
B
root(5)(x^((1)/(4)))
c
(x^(4))^(5)
D None of the above

Which expressions are equivalent to $x^{\frac{4}{5}}$ ? $\newline$ Choose all answers that apply: $\newline$ A $(\sqrt[4]{x})^{5}$ $\newline$ B $\sqrt[5]{x^{\frac{1}{4}}}$ $\newline$ C $\left(x^{4}\right)^{5}$ $\newline$ D None of the above

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Identify equivalent linear expressions

Full solution

Q. Which expressions are equivalent to

x^{\frac{4}{5}}

?

\newline

Choose all answers that apply:

\newline

A

(\sqrt[4]{x})^{5}

\newline

B

\sqrt[5]{x^{\frac{1}{4}}}

\newline

C

\left(x^{4}\right)^{5}

\newline

D None of the above

Understanding exponents: Understand the properties of exponents. The expression $x^{\frac{4}{5}}$ can be rewritten using the property of exponents that states $x^{\frac{a}{b}} = (\sqrt[b]{x})^a$ .
Applying property to option A: Apply the property to option A. $\newline$ Option A is $(\sqrt[4]{x})^5$ . According to the property, this is equivalent to $x^{\frac{5}{4}}$ , which is not the same as $x^{\frac{4}{5}}$ .
Applying property to option B: Apply the property to option B. $\newline$ Option B is $\sqrt[5]{x^{\frac{1}{4}}}$ . This can be rewritten as $(x^{\frac{1}{4}})^{\frac{1}{5}}$ , which according to the property of exponents $(x^{a})^{b} = x^{a*b}$ , is $x^{(\frac{1}{4})*(\frac{1}{5})} = x^{\frac{1}{20}}$ , which is not the same as $x^{\frac{4}{5}}$ .
Applying property to option C: Apply the property to option C. $\newline$ Option C is $(x^4)^5$ . According to the property of exponents $(x^a)^b = x^{a*b}$ , this is $x^{4*5} = x^{20}$ , which is not the same as $x^{\frac{4}{5}}$ .
Determining equivalent options: Determine if any of the options are equivalent to $x^{\frac{4}{5}}$ . None of the options $A$ , $B$ , or $C$ are equivalent to $x^{\frac{4}{5}}$ , so the correct answer is $D$ , None of the above.

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