Simplify ((3z^(1/4))^3)/(z^(1/12))

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

Step 1

Simplify the numerator.

Apply the product rule to $3 z^{\frac{1}{4}}$ .

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

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

$\frac{27 {(z^{\frac{1}{4}})}^{3}}{z^{\frac{1}{12}}}$

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

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

$\frac{27 z^{\frac{1}{4} \cdot 3}}{z^{\frac{1}{12}}}$

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

$\frac{27 z^{\frac{3}{4}}}{z^{\frac{1}{12}}}$

Step 2

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

$27 z^{\frac{3}{4}} z^{- \frac{1}{12}}$

Step 3

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

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

$27 (z^{- \frac{1}{12}} z^{\frac{3}{4}})$

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

$27 z^{- \frac{1}{12} + \frac{3}{4}}$

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

$27 z^{- \frac{1}{12} + \frac{3}{4} \cdot \frac{3}{3}}$

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

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

$27 z^{- \frac{1}{12} + \frac{3 \cdot 3}{4 \cdot 3}}$

Multiply $4$ by $3$ .

$27 z^{- \frac{1}{12} + \frac{3 \cdot 3}{12}}$

Combine the numerators over the common denominator.

$27 z^{\frac{- 1 + 3 \cdot 3}{12}}$

Simplify the numerator.

Multiply $3$ by $3$ .

$27 z^{\frac{- 1 + 9}{12}}$

Add $- 1$ and $9$ .

$27 z^{\frac{8}{12}}$

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

Factor $4$ out of $8$ .

$27 z^{\frac{4 (2)}{12}}$

Cancel the common factors.

Factor $4$ out of $12$ .

$27 z^{\frac{4 \cdot 2}{4 \cdot 3}}$

Cancel the common factor.

$27 z^{\frac{4 \cdot 2}{4 \cdot 3}}$

Rewrite the expression.

$27 z^{\frac{2}{3}}$