Answer :
To simplify the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex], we will use the property of exponents for cube roots:
[tex]\(\sqrt[3]{a} = a^{1/3}\)[/tex].
Let's break it down step by step:
1. Rewrite each cube root:
- [tex]\(\sqrt[3]{5x}\)[/tex] can be written as: [tex]\((5x)^{1/3}\)[/tex]
- [tex]\(\sqrt[3]{25x^2}\)[/tex] can be written as: [tex]\((25x^2)^{1/3}\)[/tex]
2. Multiply the expressions:
- Now, use the property of exponents that states when you multiply two expressions with the same exponent, you can combine them:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = (5x)^{1/3} \cdot (25x^2)^{1/3} = (5x \cdot 25x^2)^{1/3}
\][/tex]
3. Simplify inside the cube root:
- Multiply the terms inside the cube root:
[tex]\[
5x \cdot 25x^2 = 125x^3
\][/tex]
4. Combine and simplify:
- Now, simplify [tex]\((125x^3)^{1/3}\)[/tex]. The cube root of 125 is 5, and the cube root of [tex]\(x^3\)[/tex] is [tex]\(x\)[/tex]:
[tex]\[
(125x^3)^{1/3} = 5x
\][/tex]
Therefore, the simplified expression is [tex]\(5x\)[/tex].
[tex]\(\sqrt[3]{a} = a^{1/3}\)[/tex].
Let's break it down step by step:
1. Rewrite each cube root:
- [tex]\(\sqrt[3]{5x}\)[/tex] can be written as: [tex]\((5x)^{1/3}\)[/tex]
- [tex]\(\sqrt[3]{25x^2}\)[/tex] can be written as: [tex]\((25x^2)^{1/3}\)[/tex]
2. Multiply the expressions:
- Now, use the property of exponents that states when you multiply two expressions with the same exponent, you can combine them:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = (5x)^{1/3} \cdot (25x^2)^{1/3} = (5x \cdot 25x^2)^{1/3}
\][/tex]
3. Simplify inside the cube root:
- Multiply the terms inside the cube root:
[tex]\[
5x \cdot 25x^2 = 125x^3
\][/tex]
4. Combine and simplify:
- Now, simplify [tex]\((125x^3)^{1/3}\)[/tex]. The cube root of 125 is 5, and the cube root of [tex]\(x^3\)[/tex] is [tex]\(x\)[/tex]:
[tex]\[
(125x^3)^{1/3} = 5x
\][/tex]
Therefore, the simplified expression is [tex]\(5x\)[/tex].
