Answer :
Let's simplify the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex].
1. Use the property of cube roots:
The property of cube roots tells us that [tex]\(\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}\)[/tex].
2. Combine the expressions under a single cube root:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]
3. Multiply inside the cube root:
[tex]\[
(5x) \cdot (25x^2) = 5 \cdot 25 \cdot x \cdot x^2 = 125x^3
\][/tex]
4. Simplify the expression:
Now we have [tex]\(\sqrt[3]{125x^3}\)[/tex].
5. Calculate the cube root:
[tex]\[
\sqrt[3]{125x^3} = \sqrt[3]{125} \cdot \sqrt[3]{x^3}
\][/tex]
- The cube root of 125 is 5 because [tex]\(5 \cdot 5 \cdot 5 = 125\)[/tex].
- The cube root of [tex]\(x^3\)[/tex] is [tex]\(x\)[/tex].
6. Put it together:
[tex]\[
\sqrt[3]{125x^3} = 5 \cdot x = 5x
\][/tex]
Therefore, the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] simplifies to [tex]\(5x\)[/tex].
The correct answer is [tex]\(5x\)[/tex].
1. Use the property of cube roots:
The property of cube roots tells us that [tex]\(\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}\)[/tex].
2. Combine the expressions under a single cube root:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]
3. Multiply inside the cube root:
[tex]\[
(5x) \cdot (25x^2) = 5 \cdot 25 \cdot x \cdot x^2 = 125x^3
\][/tex]
4. Simplify the expression:
Now we have [tex]\(\sqrt[3]{125x^3}\)[/tex].
5. Calculate the cube root:
[tex]\[
\sqrt[3]{125x^3} = \sqrt[3]{125} \cdot \sqrt[3]{x^3}
\][/tex]
- The cube root of 125 is 5 because [tex]\(5 \cdot 5 \cdot 5 = 125\)[/tex].
- The cube root of [tex]\(x^3\)[/tex] is [tex]\(x\)[/tex].
6. Put it together:
[tex]\[
\sqrt[3]{125x^3} = 5 \cdot x = 5x
\][/tex]
Therefore, the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] simplifies to [tex]\(5x\)[/tex].
The correct answer is [tex]\(5x\)[/tex].
