Answer :
To simplify the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex], we need to follow these steps:
1. Understand the Problem:
We are dealing with cube roots and need to simplify the product of two cube root expressions:
[tex]\[
\sqrt[3]{5x} \text{ and } \sqrt[3]{25x^2}.
\][/tex]
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 expressions under a single cube root:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}.
\][/tex]
3. Multiply the Terms Inside the Cube Root:
Multiply [tex]\(5x\)[/tex] and [tex]\(25x^2\)[/tex]:
[tex]\[
(5x) \cdot (25x^2) = 5 \cdot 25 \cdot x \cdot x^2.
\][/tex]
Calculate the numerical multiplication:
[tex]\[
5 \cdot 25 = 125.
\][/tex]
Multiply the powers of [tex]\(x\)[/tex]:
[tex]\[
x \cdot x^2 = x^{1 + 2} = x^3.
\][/tex]
Combine these results:
[tex]\[
125x^3.
\][/tex]
4. Take the Cube Root:
Now apply the cube root to the entire expression:
[tex]\[
\sqrt[3]{125x^3}.
\][/tex]
5. Simplify the Cube Root:
Take the cube root of each part:
[tex]\[
\sqrt[3]{125} \text{ and } \sqrt[3]{x^3}.
\][/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]\((x^3)^{1/3} = x^{3/3} = x\)[/tex].
6. Combine the Simplified Terms:
Thus, the entire expression simplifies to:
[tex]\[
5x.
\][/tex]
Therefore, the fully simplified expression is [tex]\(5x\)[/tex].
1. Understand the Problem:
We are dealing with cube roots and need to simplify the product of two cube root expressions:
[tex]\[
\sqrt[3]{5x} \text{ and } \sqrt[3]{25x^2}.
\][/tex]
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 expressions under a single cube root:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}.
\][/tex]
3. Multiply the Terms Inside the Cube Root:
Multiply [tex]\(5x\)[/tex] and [tex]\(25x^2\)[/tex]:
[tex]\[
(5x) \cdot (25x^2) = 5 \cdot 25 \cdot x \cdot x^2.
\][/tex]
Calculate the numerical multiplication:
[tex]\[
5 \cdot 25 = 125.
\][/tex]
Multiply the powers of [tex]\(x\)[/tex]:
[tex]\[
x \cdot x^2 = x^{1 + 2} = x^3.
\][/tex]
Combine these results:
[tex]\[
125x^3.
\][/tex]
4. Take the Cube Root:
Now apply the cube root to the entire expression:
[tex]\[
\sqrt[3]{125x^3}.
\][/tex]
5. Simplify the Cube Root:
Take the cube root of each part:
[tex]\[
\sqrt[3]{125} \text{ and } \sqrt[3]{x^3}.
\][/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]\((x^3)^{1/3} = x^{3/3} = x\)[/tex].
6. Combine the Simplified Terms:
Thus, the entire expression simplifies to:
[tex]\[
5x.
\][/tex]
Therefore, the fully simplified expression is [tex]\(5x\)[/tex].
