Answer :
Let's simplify the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex].
### Step-by-Step Solution:
1. Use the property of cube roots:
The property of cube roots that we will use is:
[tex]\[
\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}
\][/tex]
Applying this to our expression:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]
2. Simplify the expression under the cube root:
Calculate [tex]\((5x) \cdot (25x^2)\)[/tex]:
[tex]\[
(5x) \cdot (25x^2) = 5 \cdot 25 \cdot x \cdot x^2
\][/tex]
[tex]\[
= 125 \cdot x^{1+2}
\][/tex]
[tex]\[
= 125x^3
\][/tex]
3. Take the cube root:
Now we need to compute [tex]\(\sqrt[3]{125x^3}\)[/tex]:
[tex]\[
\sqrt[3]{125x^3} = \sqrt[3]{125} \cdot \sqrt[3]{x^3}
\][/tex]
4. Simplify each part:
- 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]\((x^3)^{1/3} = x^{3/3} = x\)[/tex].
Therefore:
[tex]\[
\sqrt[3]{125} \cdot \sqrt[3]{x^3} = 5 \cdot x = 5x
\][/tex]
So, the simplified expression is [tex]\(\boxed{5x}\)[/tex].
### Step-by-Step Solution:
1. Use the property of cube roots:
The property of cube roots that we will use is:
[tex]\[
\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}
\][/tex]
Applying this to our expression:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]
2. Simplify the expression under the cube root:
Calculate [tex]\((5x) \cdot (25x^2)\)[/tex]:
[tex]\[
(5x) \cdot (25x^2) = 5 \cdot 25 \cdot x \cdot x^2
\][/tex]
[tex]\[
= 125 \cdot x^{1+2}
\][/tex]
[tex]\[
= 125x^3
\][/tex]
3. Take the cube root:
Now we need to compute [tex]\(\sqrt[3]{125x^3}\)[/tex]:
[tex]\[
\sqrt[3]{125x^3} = \sqrt[3]{125} \cdot \sqrt[3]{x^3}
\][/tex]
4. Simplify each part:
- 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]\((x^3)^{1/3} = x^{3/3} = x\)[/tex].
Therefore:
[tex]\[
\sqrt[3]{125} \cdot \sqrt[3]{x^3} = 5 \cdot x = 5x
\][/tex]
So, the simplified expression is [tex]\(\boxed{5x}\)[/tex].
