Answer :
Sure! Let's work through this problem step by step.
We are given the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] and asked to simplify it.
### Step 1: Use the Product Property of Cube Roots
The product property of cube roots states that:
[tex]\[
\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}
\][/tex]
Using this property, we can combine the cube roots:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]
### Step 2: Simplify the Expression Inside the Cube Root
Now, let's simplify the expression inside the cube root:
[tex]\[
(5x) \cdot (25x^2) = 5 \cdot 25 \cdot x \cdot x^2
\][/tex]
Calculating the numerical part:
[tex]\[
5 \cdot 25 = 125
\][/tex]
Combining the [tex]\(x\)[/tex] terms:
[tex]\[
x \cdot x^2 = x^{1+2} = x^3
\][/tex]
So the expression inside the cube root is:
[tex]\[
125x^3
\][/tex]
### Step 3: Simplify the Cube Root
Now we need to find the cube root of [tex]\(125x^3\)[/tex]:
[tex]\[
\sqrt[3]{125x^3} = \sqrt[3]{125} \cdot \sqrt[3]{x^3}
\][/tex]
#### Cube Root of 125:
Since [tex]\(125 = 5^3\)[/tex], the cube root of [tex]\(125\)[/tex] is [tex]\(5\)[/tex]:
[tex]\[
\sqrt[3]{125} = 5
\][/tex]
#### Cube Root of [tex]\(x^3\)[/tex]:
The cube root of [tex]\(x^3\)[/tex] is [tex]\(x\)[/tex]:
[tex]\[
\sqrt[3]{x^3} = x
\][/tex]
### Final Result:
Combining both parts, we get:
[tex]\[
5 \cdot x = 5x
\][/tex]
So, the simplified expression is [tex]\(\boxed{5x}\)[/tex].
We are given the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] and asked to simplify it.
### Step 1: Use the Product Property of Cube Roots
The product property of cube roots states that:
[tex]\[
\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}
\][/tex]
Using this property, we can combine the cube roots:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]
### Step 2: Simplify the Expression Inside the Cube Root
Now, let's simplify the expression inside the cube root:
[tex]\[
(5x) \cdot (25x^2) = 5 \cdot 25 \cdot x \cdot x^2
\][/tex]
Calculating the numerical part:
[tex]\[
5 \cdot 25 = 125
\][/tex]
Combining the [tex]\(x\)[/tex] terms:
[tex]\[
x \cdot x^2 = x^{1+2} = x^3
\][/tex]
So the expression inside the cube root is:
[tex]\[
125x^3
\][/tex]
### Step 3: Simplify the Cube Root
Now we need to find the cube root of [tex]\(125x^3\)[/tex]:
[tex]\[
\sqrt[3]{125x^3} = \sqrt[3]{125} \cdot \sqrt[3]{x^3}
\][/tex]
#### Cube Root of 125:
Since [tex]\(125 = 5^3\)[/tex], the cube root of [tex]\(125\)[/tex] is [tex]\(5\)[/tex]:
[tex]\[
\sqrt[3]{125} = 5
\][/tex]
#### Cube Root of [tex]\(x^3\)[/tex]:
The cube root of [tex]\(x^3\)[/tex] is [tex]\(x\)[/tex]:
[tex]\[
\sqrt[3]{x^3} = x
\][/tex]
### Final Result:
Combining both parts, we get:
[tex]\[
5 \cdot x = 5x
\][/tex]
So, the simplified expression is [tex]\(\boxed{5x}\)[/tex].
