Answer :
Sure! Let's start by simplifying the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex].
1. Combine the cube roots:
When you multiply cube roots with the same index, you can combine them under a single cube root:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]
2. Multiply the expressions inside the cube root:
Inside the cube root, multiply the coefficients and the variables separately:
[tex]\[
(5x) \cdot (25x^2) = 5 \cdot 25 \cdot x \cdot x^2
\][/tex]
3. Simplify the multiplication:
[tex]\[
5 \cdot 25 = 125
\][/tex]
[tex]\[
x \cdot x^2 = x^{1+2} = x^3
\][/tex]
So, inside the cube root, we have:
[tex]\[
125x^3
\][/tex]
4. Take the cube root:
The cube root of a product is the product of the cube roots:
[tex]\[
\sqrt[3]{125x^3} = \sqrt[3]{125} \cdot \sqrt[3]{x^3}
\][/tex]
Since [tex]\(5^3 = 125\)[/tex], [tex]\(\sqrt[3]{125} = 5\)[/tex].
Also, the cube root of [tex]\(x^3\)[/tex] is [tex]\(x\)[/tex]:
[tex]\[
\sqrt[3]{x^3} = x
\][/tex]
5. Final result:
Combine these results:
[tex]\[
\sqrt[3]{125x^3} = 5x
\][/tex]
So, the simplified expression is [tex]\(5x\)[/tex].
The answer is:
[tex]\[
\boxed{5x}
\][/tex]
1. Combine the cube roots:
When you multiply cube roots with the same index, you can combine them under a single cube root:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]
2. Multiply the expressions inside the cube root:
Inside the cube root, multiply the coefficients and the variables separately:
[tex]\[
(5x) \cdot (25x^2) = 5 \cdot 25 \cdot x \cdot x^2
\][/tex]
3. Simplify the multiplication:
[tex]\[
5 \cdot 25 = 125
\][/tex]
[tex]\[
x \cdot x^2 = x^{1+2} = x^3
\][/tex]
So, inside the cube root, we have:
[tex]\[
125x^3
\][/tex]
4. Take the cube root:
The cube root of a product is the product of the cube roots:
[tex]\[
\sqrt[3]{125x^3} = \sqrt[3]{125} \cdot \sqrt[3]{x^3}
\][/tex]
Since [tex]\(5^3 = 125\)[/tex], [tex]\(\sqrt[3]{125} = 5\)[/tex].
Also, the cube root of [tex]\(x^3\)[/tex] is [tex]\(x\)[/tex]:
[tex]\[
\sqrt[3]{x^3} = x
\][/tex]
5. Final result:
Combine these results:
[tex]\[
\sqrt[3]{125x^3} = 5x
\][/tex]
So, the simplified expression is [tex]\(5x\)[/tex].
The answer is:
[tex]\[
\boxed{5x}
\][/tex]
