Question

1. choose the answer that shows the expression in simplest form.

$left( sqrt4{625}
ight)^3$
a. 5
b. $625^{\frac{3}{4}}$
c. 125
d. $625^{\frac{4}{3}}$

Explanation:

Step1: Recall the property of radicals and exponents

The \(n\)-th root of a number \(a\) can be written as \(a^{\frac{1}{n}}\). So, \(\sqrt[4]{625}=625^{\frac{1}{4}}\).

Step2: Simplify the base

We know that \(625 = 5^4\), so substitute \(625\) with \(5^4\) in the expression: \((\sqrt[4]{625})^3=( (5^4)^{\frac{1}{4}} )^3\).

Step3: Apply the exponent rule \((a^m)^n=a^{m\times n}\)

First, for \((5^4)^{\frac{1}{4}}\), we have \(5^{4\times\frac{1}{4}} = 5^1=5\). Then, raise this result to the power of 3: \(5^3 = 125\).

Answer:

C. 125