Simplify ((2z^(1/5))^4)/(z^(1/20))

$\frac{{(2 z^{\frac{1}{5}})}^{4}}{z^{\frac{1}{20}}}$

Step 1

Simplify the numerator.

Apply the product rule to $2 z^{\frac{1}{5}}$ .

$\frac{2^{4} {(z^{\frac{1}{5}})}^{4}}{z^{\frac{1}{20}}}$

Raise $2$ to the power of $4$ .

$\frac{16 {(z^{\frac{1}{5}})}^{4}}{z^{\frac{1}{20}}}$

Multiply the exponents in ${(z^{\frac{1}{5}})}^{4}$ .

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

$\frac{16 z^{\frac{1}{5} \cdot 4}}{z^{\frac{1}{20}}}$

Combine $\frac{1}{5}$ and $4$ .

$\frac{16 z^{\frac{4}{5}}}{z^{\frac{1}{20}}}$

Step 2

Move $z^{\frac{1}{20}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$16 z^{\frac{4}{5}} z^{- \frac{1}{20}}$

Step 3

Multiply $z^{\frac{4}{5}}$ by $z^{- \frac{1}{20}}$ by adding the exponents.

Move $z^{- \frac{1}{20}}$ .

$16 (z^{- \frac{1}{20}} z^{\frac{4}{5}})$

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

$16 z^{- \frac{1}{20} + \frac{4}{5}}$

To write $\frac{4}{5}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$16 z^{- \frac{1}{20} + \frac{4}{5} \cdot \frac{4}{4}}$

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{4}{5}$ by $\frac{4}{4}$ .

$16 z^{- \frac{1}{20} + \frac{4 \cdot 4}{5 \cdot 4}}$

Multiply $5$ by $4$ .

$16 z^{- \frac{1}{20} + \frac{4 \cdot 4}{20}}$

Combine the numerators over the common denominator.

$16 z^{\frac{- 1 + 4 \cdot 4}{20}}$

Simplify the numerator.

Multiply $4$ by $4$ .

$16 z^{\frac{- 1 + 16}{20}}$

Add $- 1$ and $16$ .

$16 z^{\frac{15}{20}}$

Cancel the common factor of $15$ and $20$ .

Factor $5$ out of $15$ .

$16 z^{\frac{5 (3)}{20}}$

Cancel the common factors.

Factor $5$ out of $20$ .

$16 z^{\frac{5 \cdot 3}{5 \cdot 4}}$

Cancel the common factor.

$16 z^{\frac{5 \cdot 3}{5 \cdot 4}}$

Rewrite the expression.

$16 z^{\frac{3}{4}}$