Simplify ((2x^(1/2))^3)/(4x^2)

$\frac{{(2 x^{\frac{1}{2}})}^{3}}{4 x^{2}}$

Step 1

Simplify the numerator.

Apply the product rule to $2 x^{\frac{1}{2}}$ .

$\frac{2^{3} {(x^{\frac{1}{2}})}^{3}}{4 x^{2}}$

Raise $2$ to the power of $3$ .

$\frac{8 {(x^{\frac{1}{2}})}^{3}}{4 x^{2}}$

Multiply the exponents in ${(x^{\frac{1}{2}})}^{3}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{8 x^{\frac{1}{2} \cdot 3}}{4 x^{2}}$

Combine $\frac{1}{2}$ and $3$ .

$\frac{8 x^{\frac{3}{2}}}{4 x^{2}}$

Step 2

Move $x^{\frac{3}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{8}{4 x^{2} x^{- \frac{3}{2}}}$

Step 3

Multiply $x^{2}$ by $x^{- \frac{3}{2}}$ by adding the exponents.

Move $x^{- \frac{3}{2}}$ .

$\frac{8}{4 (x^{- \frac{3}{2}} x^{2})}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{8}{4 x^{- \frac{3}{2} + 2}}$

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{8}{4 x^{- \frac{3}{2} + 2 \cdot \frac{2}{2}}}$

Combine $2$ and $\frac{2}{2}$ .

$\frac{8}{4 x^{- \frac{3}{2} + \frac{2 \cdot 2}{2}}}$

Combine the numerators over the common denominator.

$\frac{8}{4 x^{\frac{- 3 + 2 \cdot 2}{2}}}$

Simplify the numerator.

Multiply $2$ by $2$ .

$\frac{8}{4 x^{\frac{- 3 + 4}{2}}}$

Add $- 3$ and $4$ .

$\frac{8}{4 x^{\frac{1}{2}}}$

Step 4

Factor $4$ out of $8$ .

$\frac{4 \cdot 2}{4 x^{\frac{1}{2}}}$

Step 5

Cancel the common factors.

Factor $4$ out of $4 x^{\frac{1}{2}}$ .

$\frac{4 \cdot 2}{4 (x^{\frac{1}{2}})}$

Cancel the common factor.

$\frac{4 \cdot 2}{4 x^{\frac{1}{2}}}$

Rewrite the expression.

$\frac{2}{x^{\frac{1}{2}}}$