To simplify the expression [tex]\(125^{\frac{2}{3}}\)[/tex], we can follow these steps:
1. Express 125 as a power of a smaller number:
[tex]\(125\)[/tex] can be written as [tex]\(5^3\)[/tex] because [tex]\(125 = 5 \times 5 \times 5\)[/tex].
2. Rewrite the given expression:
Substitute [tex]\(125\)[/tex] with [tex]\(5^3\)[/tex]:
[tex]\[
125^{\frac{2}{3}} = (5^3)^{\frac{2}{3}}
\][/tex]
3. Use the properties of exponents:
Recall that [tex]\((a^m)^n = a^{m \cdot n}\)[/tex].
Apply this property:
[tex]\[
(5^3)^{\frac{2}{3}} = 5^{3 \cdot \frac{2}{3}}
\][/tex]
4. Simplify the exponent:
Multiply the exponents:
[tex]\[
5^{3 \cdot \frac{2}{3}} = 5^2
\][/tex]
5. Calculate [tex]\(5^2\)[/tex]:
[tex]\[
5^2 = 25
\][/tex]
Therefore, the simplified form of the expression [tex]\(125^{\frac{2}{3}}\)[/tex] is:
[tex]\[
\boxed{25}
\][/tex]
