Answer :
To simplify the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex], we follow these steps:
1. Understand the expression: We have two cube roots, [tex]\(\sqrt[3]{5x}\)[/tex] and [tex]\(\sqrt[3]{25x^2}\)[/tex], that need to be multiplied together.
2. Use the property of cube roots: The property [tex]\(\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}\)[/tex] allows us to combine the two cube roots into one. So, we rewrite the expression as:
[tex]\[
\sqrt[3]{5x \cdot 25x^2}
\][/tex]
3. Multiply inside the cube root: Inside the cube root, multiply the terms:
- Multiply the numbers: [tex]\(5 \cdot 25 = 125\)[/tex]
- Multiply the [tex]\(x\)[/tex] terms: [tex]\(x \cdot x^2 = x^3\)[/tex]
This gives us:
[tex]\[
\sqrt[3]{125x^3}
\][/tex]
4. Simplify the cube root: Now, simplify [tex]\(\sqrt[3]{125x^3}\)[/tex]:
- The cube root of 125 is 5, because [tex]\(5^3 = 125\)[/tex].
- The cube root of [tex]\(x^3\)[/tex] is [tex]\(x\)[/tex], because [tex]\((x^3)^{1/3} = x\)[/tex].
5. Final result: Therefore, the simplified expression is:
[tex]\[
5x
\][/tex]
So, the answer is [tex]\(5x\)[/tex].
1. Understand the expression: We have two cube roots, [tex]\(\sqrt[3]{5x}\)[/tex] and [tex]\(\sqrt[3]{25x^2}\)[/tex], that need to be multiplied together.
2. Use the property of cube roots: The property [tex]\(\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}\)[/tex] allows us to combine the two cube roots into one. So, we rewrite the expression as:
[tex]\[
\sqrt[3]{5x \cdot 25x^2}
\][/tex]
3. Multiply inside the cube root: Inside the cube root, multiply the terms:
- Multiply the numbers: [tex]\(5 \cdot 25 = 125\)[/tex]
- Multiply the [tex]\(x\)[/tex] terms: [tex]\(x \cdot x^2 = x^3\)[/tex]
This gives us:
[tex]\[
\sqrt[3]{125x^3}
\][/tex]
4. Simplify the cube root: Now, simplify [tex]\(\sqrt[3]{125x^3}\)[/tex]:
- The cube root of 125 is 5, because [tex]\(5^3 = 125\)[/tex].
- The cube root of [tex]\(x^3\)[/tex] is [tex]\(x\)[/tex], because [tex]\((x^3)^{1/3} = x\)[/tex].
5. Final result: Therefore, the simplified expression is:
[tex]\[
5x
\][/tex]
So, the answer is [tex]\(5x\)[/tex].
