Simplify ((a^(3/5)b^(1/2))/(a^(1/5)))^(1/4)

${(\frac{a^{\frac{3}{5}} b^{\frac{1}{2}}}{a^{\frac{1}{5}}})}^{\frac{1}{4}}$

Step 1

Move $a^{\frac{1}{5}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

${(a^{\frac{3}{5}} b^{\frac{1}{2}} a^{- \frac{1}{5}})}^{\frac{1}{4}}$

Step 2

Multiply $a^{\frac{3}{5}}$ by $a^{- \frac{1}{5}}$ by adding the exponents.

Move $a^{- \frac{1}{5}}$ .

${(a^{- \frac{1}{5}} a^{\frac{3}{5}} b^{\frac{1}{2}})}^{\frac{1}{4}}$

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

${(a^{- \frac{1}{5} + \frac{3}{5}} b^{\frac{1}{2}})}^{\frac{1}{4}}$

Combine the numerators over the common denominator.

${(a^{\frac{- 1 + 3}{5}} b^{\frac{1}{2}})}^{\frac{1}{4}}$

Add $- 1$ and $3$ .

${(a^{\frac{2}{5}} b^{\frac{1}{2}})}^{\frac{1}{4}}$

Step 3

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

${(a^{\frac{2}{5}})}^{\frac{1}{4}} {(b^{\frac{1}{2}})}^{\frac{1}{4}}$

Step 4

Multiply the exponents in ${(a^{\frac{2}{5}})}^{\frac{1}{4}}$ .

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

$a^{\frac{2}{5} \cdot \frac{1}{4}} {(b^{\frac{1}{2}})}^{\frac{1}{4}}$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$a^{\frac{2}{5} \cdot \frac{1}{2 (2)}} {(b^{\frac{1}{2}})}^{\frac{1}{4}}$

Cancel the common factor.

$a^{\frac{2}{5} \cdot \frac{1}{2 \cdot 2}} {(b^{\frac{1}{2}})}^{\frac{1}{4}}$

Rewrite the expression.

$a^{\frac{1}{5} \cdot \frac{1}{2}} {(b^{\frac{1}{2}})}^{\frac{1}{4}}$

Multiply $\frac{1}{5}$ by $\frac{1}{2}$ .

$a^{\frac{1}{5 \cdot 2}} {(b^{\frac{1}{2}})}^{\frac{1}{4}}$

Multiply $5$ by $2$ .

$a^{\frac{1}{10}} {(b^{\frac{1}{2}})}^{\frac{1}{4}}$

Step 5

Multiply the exponents in ${(b^{\frac{1}{2}})}^{\frac{1}{4}}$ .

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

$a^{\frac{1}{10}} b^{\frac{1}{2} \cdot \frac{1}{4}}$

Multiply $\frac{1}{2} \cdot \frac{1}{4}$ .

Multiply $\frac{1}{2}$ by $\frac{1}{4}$ .

$a^{\frac{1}{10}} b^{\frac{1}{2 \cdot 4}}$

Multiply $2$ by $4$ .

$a^{\frac{1}{10}} b^{\frac{1}{8}}$