To determine the correct answer to the expression [tex]\((625 \cdot 48)^{\frac{1}{4}}\)[/tex], let's work through it step by step.
First, let's break down the given expression [tex]\((625 \cdot 48)^{\frac{1}{4}}\)[/tex].
1. Evaluate the expression inside the parentheses:
[tex]\[
625 \cdot 48
\][/tex]
2. Using factorization:
- [tex]\(625 = 5^4\)[/tex] because [tex]\(625 = 5 \cdot 5 \cdot 5 \cdot 5\)[/tex]
- [tex]\(48 = 16 \cdot 3 = 2^4 \cdot 3\)[/tex]
3. Combine these factors:
[tex]\[
625 \cdot 48 = 5^4 \cdot (2^4 \cdot 3) = (5 \cdot 2)^4 \cdot 3 = 10^4 \cdot 3
\][/tex]
4. Now apply the fourth root:
[tex]\[
(10^4 \cdot 3)^{\frac{1}{4}} = 10^{4 \cdot (1/4)} \cdot 3^{1/4} = 10^1 \cdot 3^{1/4} = 10 \cdot 3^{1/4}
\][/tex]
Therefore, we have:
[tex]\[
(625 \cdot 48)^{\frac{1}{4}} = 10 \sqrt[4]{3}
\][/tex]
Given the provided options:
- 60
- 30
- [tex]\(40 \sqrt[4]{3}\)[/tex]
- [tex]\(10 \sqrt[4]{3}\)[/tex]
The correct answer is:
[tex]\[
10 \sqrt[4]{3}
\][/tex]
So, the expression [tex]\((625 \cdot 48)^{\frac{1}{4}}\)[/tex] equals [tex]\(10 \sqrt[4]{3}\)[/tex].
