Certainly! Let's simplify the given expression step by step:
The original expression is:
[tex]\[
\frac{(2x \cdot x^{1/2})^3}{4x^2}
\][/tex]
### Step 1: Simplify the numerator
First, consider the inside of the numerator:
[tex]\[
2x \cdot x^{1/2}
\][/tex]
We can multiply the terms with the same base (i.e., [tex]\(x\)[/tex]) by adding their exponents:
[tex]\[
2x \cdot x^{1/2} = 2x^{1 + 1/2} = 2x^{3/2}
\][/tex]
Now, raise this result to the power of 3:
[tex]\[
(2x^{3/2})^3
\][/tex]
Apply the exponentiation rules, where you distribute the exponent across the product and multiply the exponents:
[tex]\[
(2^3) \cdot (x^{3/2})^3 = 8 \cdot x^{(3/2) \cdot 3} = 8x^{9/2}
\][/tex]
So, the numerator simplifies to:
[tex]\[
8x^{9/2}
\][/tex]
### Step 2: Simplify the denominator
The denominator is:
[tex]\[
4x^2
\][/tex]
### Step 3: Combine the simplified numerator and denominator
Now we have the expression:
[tex]\[
\frac{8x^{9/2}}{4x^2}
\][/tex]
### Step 4: Simplify the fraction
Divide both the coefficient and the powers of [tex]\(x\)[/tex]:
[tex]\[
\frac{8}{4} \cdot \frac{x^{9/2}}{x^2} = 2 \cdot x^{(9/2 - 2)}
\][/tex]
Simplify the exponent of [tex]\(x\)[/tex]:
[tex]\[
9/2 - 2 = 9/2 - 4/2 = 5/2
\][/tex]
So, the expression becomes:
[tex]\[
2x^{5/2}
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
2x^{5/2}
\][/tex]
