Answer :
To simplify the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex], let's follow these steps:
1. Understand the Product Rule for Roots:
When you multiply two cube roots, you can combine them into a single cube root, like this:
[tex]\[
\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{ab}
\][/tex]
2. Apply the Product Rule:
Using the product rule, we can combine the two cube roots in the expression:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]
3. Multiply Inside the Cube Root:
Now, multiply the expressions inside the cube root:
[tex]\[
(5x) \cdot (25x^2) = 125x^3
\][/tex]
4. Simplify the Cube Root:
Take the cube root of [tex]\(125x^3\)[/tex]:
[tex]\[
\sqrt[3]{125x^3} = \sqrt[3]{125} \cdot \sqrt[3]{x^3}
\][/tex]
- The cube root of 125 is 5, because [tex]\(5 \times 5 \times 5 = 125\)[/tex].
- The cube root of [tex]\(x^3\)[/tex] is [tex]\(x\)[/tex], because [tex]\((x^3)^{1/3} = x\)[/tex].
5. Combine the Results:
Putting it all together, we have:
[tex]\[
\sqrt[3]{125} \cdot \sqrt[3]{x^3} = 5x
\][/tex]
So, the simplified form of [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] is [tex]\(5x\)[/tex].
1. Understand the Product Rule for Roots:
When you multiply two cube roots, you can combine them into a single cube root, like this:
[tex]\[
\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{ab}
\][/tex]
2. Apply the Product Rule:
Using the product rule, we can combine the two cube roots in the expression:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]
3. Multiply Inside the Cube Root:
Now, multiply the expressions inside the cube root:
[tex]\[
(5x) \cdot (25x^2) = 125x^3
\][/tex]
4. Simplify the Cube Root:
Take the cube root of [tex]\(125x^3\)[/tex]:
[tex]\[
\sqrt[3]{125x^3} = \sqrt[3]{125} \cdot \sqrt[3]{x^3}
\][/tex]
- The cube root of 125 is 5, because [tex]\(5 \times 5 \times 5 = 125\)[/tex].
- The cube root of [tex]\(x^3\)[/tex] is [tex]\(x\)[/tex], because [tex]\((x^3)^{1/3} = x\)[/tex].
5. Combine the Results:
Putting it all together, we have:
[tex]\[
\sqrt[3]{125} \cdot \sqrt[3]{x^3} = 5x
\][/tex]
So, the simplified form of [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] is [tex]\(5x\)[/tex].
