Answer :
Let's simplify the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] step by step.
### Step 1: Apply the property of radicals
We use the property that states [tex]\(\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}\)[/tex]. Here, both are cube roots ([tex]\(\sqrt[3]{}\)[/tex]), so we can combine them:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]
### Step 2: Simplify the expression under the cube root
First, calculate the product inside the cube root:
- Multiply the numbers: [tex]\(5 \times 25 = 125\)[/tex]
- Multiply the variables: [tex]\(x \times x^2 = x^{1+2} = x^3\)[/tex]
Now, combine them under the cube root:
[tex]\[
\sqrt[3]{125x^3}
\][/tex]
### Step 3: Simplify the cube root
Since [tex]\(125 = 5^3\)[/tex], we write the expression as:
[tex]\[
\sqrt[3]{(5^3) \cdot (x^3)}
\][/tex]
Taking the cube root of each part, we get:
[tex]\[
5 \cdot x = 5x
\][/tex]
### Conclusion
The completely simplified expression is [tex]\(5x\)[/tex].
### Step 1: Apply the property of radicals
We use the property that states [tex]\(\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}\)[/tex]. Here, both are cube roots ([tex]\(\sqrt[3]{}\)[/tex]), so we can combine them:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]
### Step 2: Simplify the expression under the cube root
First, calculate the product inside the cube root:
- Multiply the numbers: [tex]\(5 \times 25 = 125\)[/tex]
- Multiply the variables: [tex]\(x \times x^2 = x^{1+2} = x^3\)[/tex]
Now, combine them under the cube root:
[tex]\[
\sqrt[3]{125x^3}
\][/tex]
### Step 3: Simplify the cube root
Since [tex]\(125 = 5^3\)[/tex], we write the expression as:
[tex]\[
\sqrt[3]{(5^3) \cdot (x^3)}
\][/tex]
Taking the cube root of each part, we get:
[tex]\[
5 \cdot x = 5x
\][/tex]
### Conclusion
The completely simplified expression is [tex]\(5x\)[/tex].
