Answer :
To simplify the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex], we'll use properties of cube roots and exponents. Let's go through the steps:
1. Recognize the Cube Root Property:
- The property of cube roots states that [tex]\(\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}\)[/tex].
2. Multiply Underneath the Cube Roots:
- Using the cube root property, combine the expressions:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]
3. Simplify Inside the Cube Root:
- Multiply the numbers and the variables separately:
- For numbers: [tex]\(5 \times 25 = 125\)[/tex]
- For variables: [tex]\(x \times x^2 = x^{1+2} = x^3\)[/tex]
- So, the expression inside the cube root becomes:
[tex]\[
\sqrt[3]{125x^3}
\][/tex]
4. Simplify the Cube Root:
- Break it down:
- The cube root of [tex]\(125\)[/tex] is [tex]\(5\)[/tex] because [tex]\(5^3 = 125\)[/tex].
- The cube root of [tex]\(x^3\)[/tex] is [tex]\(x\)[/tex] because [tex]\((x^3)^{1/3} = x^{3/3} = x\)[/tex].
5. Combine the Simplified Parts:
- Combine the simplified parts from the cube root:
[tex]\[
5 \cdot x = 5x
\][/tex]
So, the simplified expression is [tex]\(5x\)[/tex].
The correct answer is [tex]\(\boxed{5x}\)[/tex].
1. Recognize the Cube Root Property:
- The property of cube roots states that [tex]\(\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}\)[/tex].
2. Multiply Underneath the Cube Roots:
- Using the cube root property, combine the expressions:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]
3. Simplify Inside the Cube Root:
- Multiply the numbers and the variables separately:
- For numbers: [tex]\(5 \times 25 = 125\)[/tex]
- For variables: [tex]\(x \times x^2 = x^{1+2} = x^3\)[/tex]
- So, the expression inside the cube root becomes:
[tex]\[
\sqrt[3]{125x^3}
\][/tex]
4. Simplify the Cube Root:
- Break it down:
- The cube root of [tex]\(125\)[/tex] is [tex]\(5\)[/tex] because [tex]\(5^3 = 125\)[/tex].
- The cube root of [tex]\(x^3\)[/tex] is [tex]\(x\)[/tex] because [tex]\((x^3)^{1/3} = x^{3/3} = x\)[/tex].
5. Combine the Simplified Parts:
- Combine the simplified parts from the cube root:
[tex]\[
5 \cdot x = 5x
\][/tex]
So, the simplified expression is [tex]\(5x\)[/tex].
The correct answer is [tex]\(\boxed{5x}\)[/tex].
