Answer :
Sure! Let's simplify the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] step-by-step.
1. Understand the Cubic Roots:
- The expression [tex]\(\sqrt[3]{5x}\)[/tex] can be written as [tex]\((5x)^{1/3}\)[/tex].
- The expression [tex]\(\sqrt[3]{25x^2}\)[/tex] can be written as [tex]\((25x^2)^{1/3}\)[/tex].
2. Combine the Expressions Using Properties of Exponents:
- When you multiply powers with the same root, you can combine them as a single power:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = ((5x) \cdot (25x^2))^{1/3}
\][/tex]
3. Simplify Inside the Parenthesis:
- Multiply the terms inside the parenthesis:
[tex]\[
5x \cdot 25x^2 = 125x^3
\][/tex]
4. Apply the Cubic Root:
- Now, simplify [tex]\(\sqrt[3]{125x^3}\)[/tex]:
[tex]\[
(125x^3)^{1/3}
\][/tex]
5. Break it Down:
- Break it into two parts: [tex]\((125)^{1/3}\)[/tex] and [tex]\((x^3)^{1/3}\)[/tex].
- Calculate [tex]\((125)^{1/3}\)[/tex]: Since [tex]\(125 = 5^3\)[/tex], [tex]\((125)^{1/3} = 5\)[/tex].
- Simplify [tex]\((x^3)^{1/3}\)[/tex]: This equals [tex]\(x^{3/3} = x^1 = x\)[/tex].
6. Conclude the Simplification:
- Combine the results:
[tex]\[
5 \cdot x = 5x
\][/tex]
So, the simplified form of the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] is [tex]\(5x\)[/tex].
1. Understand the Cubic Roots:
- The expression [tex]\(\sqrt[3]{5x}\)[/tex] can be written as [tex]\((5x)^{1/3}\)[/tex].
- The expression [tex]\(\sqrt[3]{25x^2}\)[/tex] can be written as [tex]\((25x^2)^{1/3}\)[/tex].
2. Combine the Expressions Using Properties of Exponents:
- When you multiply powers with the same root, you can combine them as a single power:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = ((5x) \cdot (25x^2))^{1/3}
\][/tex]
3. Simplify Inside the Parenthesis:
- Multiply the terms inside the parenthesis:
[tex]\[
5x \cdot 25x^2 = 125x^3
\][/tex]
4. Apply the Cubic Root:
- Now, simplify [tex]\(\sqrt[3]{125x^3}\)[/tex]:
[tex]\[
(125x^3)^{1/3}
\][/tex]
5. Break it Down:
- Break it into two parts: [tex]\((125)^{1/3}\)[/tex] and [tex]\((x^3)^{1/3}\)[/tex].
- Calculate [tex]\((125)^{1/3}\)[/tex]: Since [tex]\(125 = 5^3\)[/tex], [tex]\((125)^{1/3} = 5\)[/tex].
- Simplify [tex]\((x^3)^{1/3}\)[/tex]: This equals [tex]\(x^{3/3} = x^1 = x\)[/tex].
6. Conclude the Simplification:
- Combine the results:
[tex]\[
5 \cdot x = 5x
\][/tex]
So, the simplified form of the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] is [tex]\(5x\)[/tex].
