Answer :
Sure, let's go through a step-by-step solution to simplify the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex].
### Step 1: Apply the Property of Cube Roots
There's a property of roots that can be very helpful:
[tex]\[
\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}
\][/tex]
Using this property, we can combine the two 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
Multiply the expressions inside the cube root:
[tex]\[
(5x) \cdot (25x^2) = 5 \cdot 25 \cdot x \cdot x^2 = 125x^3
\][/tex]
### Step 3: Take the Cube Root of the Resulting Expression
Now, find the cube root of the expression:
[tex]\[
\sqrt[3]{125x^3}
\][/tex]
The cube root can be split into two parts:
[tex]\[
\sqrt[3]{125} \cdot \sqrt[3]{x^3}
\][/tex]
- The cube root of 125 is 5, since [tex]\(5^3 = 125\)[/tex].
- The cube root of [tex]\(x^3\)[/tex] is [tex]\(x\)[/tex], since raising [tex]\(x\)[/tex] to the power of 3 and then taking the cube root gives [tex]\(x\)[/tex].
Combine these results:
[tex]\[
\sqrt[3]{125x^3} = 5x
\][/tex]
Therefore, the simplified form of the original expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] is [tex]\(5x\)[/tex].
The correct answer is [tex]\(5x\)[/tex].
### Step 1: Apply the Property of Cube Roots
There's a property of roots that can be very helpful:
[tex]\[
\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}
\][/tex]
Using this property, we can combine the two 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
Multiply the expressions inside the cube root:
[tex]\[
(5x) \cdot (25x^2) = 5 \cdot 25 \cdot x \cdot x^2 = 125x^3
\][/tex]
### Step 3: Take the Cube Root of the Resulting Expression
Now, find the cube root of the expression:
[tex]\[
\sqrt[3]{125x^3}
\][/tex]
The cube root can be split into two parts:
[tex]\[
\sqrt[3]{125} \cdot \sqrt[3]{x^3}
\][/tex]
- The cube root of 125 is 5, since [tex]\(5^3 = 125\)[/tex].
- The cube root of [tex]\(x^3\)[/tex] is [tex]\(x\)[/tex], since raising [tex]\(x\)[/tex] to the power of 3 and then taking the cube root gives [tex]\(x\)[/tex].
Combine these results:
[tex]\[
\sqrt[3]{125x^3} = 5x
\][/tex]
Therefore, the simplified form of the original expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] is [tex]\(5x\)[/tex].
The correct answer is [tex]\(5x\)[/tex].
