Simplify ((3y^(1/5))^4)/(y^(1/20))

$\frac{{(3 y^{\frac{1}{5}})}^{4}}{y^{\frac{1}{20}}}$

Step 1

Simplify the numerator.

Apply the product rule to $3 y^{\frac{1}{5}}$ .

$\frac{3^{4} {(y^{\frac{1}{5}})}^{4}}{y^{\frac{1}{20}}}$

Raise $3$ to the power of $4$ .

$\frac{81 {(y^{\frac{1}{5}})}^{4}}{y^{\frac{1}{20}}}$

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

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

$\frac{81 y^{\frac{1}{5} \cdot 4}}{y^{\frac{1}{20}}}$

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

$\frac{81 y^{\frac{4}{5}}}{y^{\frac{1}{20}}}$

Step 2

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

$81 y^{\frac{4}{5}} y^{- \frac{1}{20}}$

Step 3

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

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

$81 (y^{- \frac{1}{20}} y^{\frac{4}{5}})$

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

$81 y^{- \frac{1}{20} + \frac{4}{5}}$

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

$81 y^{- \frac{1}{20} + \frac{4}{5} \cdot \frac{4}{4}}$

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

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

$81 y^{- \frac{1}{20} + \frac{4 \cdot 4}{5 \cdot 4}}$

Multiply $5$ by $4$ .

$81 y^{- \frac{1}{20} + \frac{4 \cdot 4}{20}}$

Combine the numerators over the common denominator.

$81 y^{\frac{- 1 + 4 \cdot 4}{20}}$

Simplify the numerator.

Multiply $4$ by $4$ .

$81 y^{\frac{- 1 + 16}{20}}$

Add $- 1$ and $16$ .

$81 y^{\frac{15}{20}}$

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

Factor $5$ out of $15$ .

$81 y^{\frac{5 (3)}{20}}$

Cancel the common factors.

Factor $5$ out of $20$ .

$81 y^{\frac{5 \cdot 3}{5 \cdot 4}}$

Cancel the common factor.

$81 y^{\frac{5 \cdot 3}{5 \cdot 4}}$

Rewrite the expression.

$81 y^{\frac{3}{4}}$