Answer :
To simplify the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex], we can follow these steps:
1. Understand the Cubic Root:
[tex]\(\sqrt[3]{a}\)[/tex] is the same as [tex]\(a^{1/3}\)[/tex]. This means each cube root expression can be rewritten in terms of exponents.
2. Rewrite the Expression:
[tex]\[
\sqrt[3]{5x} = (5x)^{1/3}
\][/tex]
[tex]\[
\sqrt[3]{25x^2} = (25x^2)^{1/3}
\][/tex]
3. Multiply the Two Expressions:
By multiplying the two cube root expressions, we get:
[tex]\[
(5x)^{1/3} \cdot (25x^2)^{1/3} = (5x \cdot 25x^2)^{1/3}
\][/tex]
4. Simplify Inside the Parentheses:
Multiply the terms inside:
[tex]\[
5x \cdot 25x^2 = 125x^3
\][/tex]
5. Simplify the Cube Root of the Product:
[tex]\[
\sqrt[3]{125x^3}
\][/tex]
6. Calculate the Cube Root:
Since [tex]\(125 = 5^3\)[/tex], we have:
[tex]\[
\sqrt[3]{125x^3} = \sqrt[3]{5^3 \cdot x^3}
\][/tex]
7. Apply the Property of Cube Roots:
[tex]\[
\sqrt[3]{5^3 \cdot x^3} = 5 \cdot x
\][/tex]
Thus, the simplified expression is [tex]\(5x\)[/tex].
1. Understand the Cubic Root:
[tex]\(\sqrt[3]{a}\)[/tex] is the same as [tex]\(a^{1/3}\)[/tex]. This means each cube root expression can be rewritten in terms of exponents.
2. Rewrite the Expression:
[tex]\[
\sqrt[3]{5x} = (5x)^{1/3}
\][/tex]
[tex]\[
\sqrt[3]{25x^2} = (25x^2)^{1/3}
\][/tex]
3. Multiply the Two Expressions:
By multiplying the two cube root expressions, we get:
[tex]\[
(5x)^{1/3} \cdot (25x^2)^{1/3} = (5x \cdot 25x^2)^{1/3}
\][/tex]
4. Simplify Inside the Parentheses:
Multiply the terms inside:
[tex]\[
5x \cdot 25x^2 = 125x^3
\][/tex]
5. Simplify the Cube Root of the Product:
[tex]\[
\sqrt[3]{125x^3}
\][/tex]
6. Calculate the Cube Root:
Since [tex]\(125 = 5^3\)[/tex], we have:
[tex]\[
\sqrt[3]{125x^3} = \sqrt[3]{5^3 \cdot x^3}
\][/tex]
7. Apply the Property of Cube Roots:
[tex]\[
\sqrt[3]{5^3 \cdot x^3} = 5 \cdot x
\][/tex]
Thus, the simplified expression is [tex]\(5x\)[/tex].
