Simplify ((2x^(1/4))^3)/(x^(1/12))

$\frac{{(2 x^{\frac{1}{4}})}^{3}}{x^{\frac{1}{12}}}$

Step 1

Simplify the numerator.

Apply the product rule to $2 x^{\frac{1}{4}}$ .

$\frac{2^{3} {(x^{\frac{1}{4}})}^{3}}{x^{\frac{1}{12}}}$

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

$\frac{8 {(x^{\frac{1}{4}})}^{3}}{x^{\frac{1}{12}}}$

Multiply the exponents in ${(x^{\frac{1}{4}})}^{3}$ .

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

$\frac{8 x^{\frac{1}{4} \cdot 3}}{x^{\frac{1}{12}}}$

Combine $\frac{1}{4}$ and $3$ .

$\frac{8 x^{\frac{3}{4}}}{x^{\frac{1}{12}}}$

Step 2

Move $x^{\frac{1}{12}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$8 x^{\frac{3}{4}} x^{- \frac{1}{12}}$

Step 3

Multiply $x^{\frac{3}{4}}$ by $x^{- \frac{1}{12}}$ by adding the exponents.

Move $x^{- \frac{1}{12}}$ .

$8 (x^{- \frac{1}{12}} x^{\frac{3}{4}})$

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

$8 x^{- \frac{1}{12} + \frac{3}{4}}$

To write $\frac{3}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$8 x^{- \frac{1}{12} + \frac{3}{4} \cdot \frac{3}{3}}$

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3}{4}$ by $\frac{3}{3}$ .

$8 x^{- \frac{1}{12} + \frac{3 \cdot 3}{4 \cdot 3}}$

Multiply $4$ by $3$ .

$8 x^{- \frac{1}{12} + \frac{3 \cdot 3}{12}}$

Combine the numerators over the common denominator.

$8 x^{\frac{- 1 + 3 \cdot 3}{12}}$

Simplify the numerator.

Multiply $3$ by $3$ .

$8 x^{\frac{- 1 + 9}{12}}$

Add $- 1$ and $9$ .

$8 x^{\frac{8}{12}}$

Cancel the common factor of $8$ and $12$ .

Factor $4$ out of $8$ .

$8 x^{\frac{4 (2)}{12}}$

Cancel the common factors.

Factor $4$ out of $12$ .

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

Cancel the common factor.

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

Rewrite the expression.

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