Simplify ( fourth root of x^9y^8)/(4x^-1y^(-5/4))

$\frac{\sqrt[4]{x^{9} y^{8}}}{4 x^{- 1} y^{- \frac{5}{4}}}$

Step 1

Simplify the expression.

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

$\frac{\sqrt[4]{x^{9} y^{8}} x}{4 y^{- \frac{5}{4}}}$

Move $y^{- \frac{5}{4}}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{\sqrt[4]{x^{9} y^{8}} x y^{\frac{5}{4}}}{4}$

Step 2

Simplify the numerator.

Rewrite $x^{9} y^{8}$ as ${(x^{2} y^{2})}^{4} x$ .

Factor out $x^{8}$ .

$\frac{\sqrt[4]{x^{8} x y^{8}} x y^{\frac{5}{4}}}{4}$

Rewrite $x^{8}$ as ${(x^{2})}^{4}$ .

$\frac{\sqrt[4]{{(x^{2})}^{4} x y^{8}} x y^{\frac{5}{4}}}{4}$

Rewrite $y^{8}$ as ${(y^{2})}^{4}$ .

$\frac{\sqrt[4]{{(x^{2})}^{4} x {(y^{2})}^{4}} x y^{\frac{5}{4}}}{4}$

Move $x$ .

$\frac{\sqrt[4]{{(x^{2})}^{4} {(y^{2})}^{4} x} x y^{\frac{5}{4}}}{4}$

Rewrite ${(x^{2})}^{4} {(y^{2})}^{4}$ as ${(x^{2} y^{2})}^{4}$ .

$\frac{\sqrt[4]{{(x^{2} y^{2})}^{4} x} x y^{\frac{5}{4}}}{4}$

Pull terms out from under the radical.

$\frac{x^{2} y^{2} \sqrt[4]{x} x y^{\frac{5}{4}}}{4}$

Combine exponents.

Raise $x$ to the power of $1$ .

$\frac{x^{1} x^{2} y^{2} \sqrt[4]{x} y^{\frac{5}{4}}}{4}$

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

$\frac{x^{1 + 2} y^{2} \sqrt[4]{x} y^{\frac{5}{4}}}{4}$

Add $1$ and $2$ .

$\frac{x^{3} y^{2} \sqrt[4]{x} y^{\frac{5}{4}}}{4}$

Multiply $y^{2}$ by $y^{\frac{5}{4}}$ by adding the exponents.

Move $y^{\frac{5}{4}}$ .

$\frac{x^{3} (y^{\frac{5}{4}} y^{2}) \sqrt[4]{x}}{4}$

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

$\frac{x^{3} y^{\frac{5}{4} + 2} \sqrt[4]{x}}{4}$

To write $2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{x^{3} y^{\frac{5}{4} + 2 \cdot \frac{4}{4}} \sqrt[4]{x}}{4}$

Combine $2$ and $\frac{4}{4}$ .

$\frac{x^{3} y^{\frac{5}{4} + \frac{2 \cdot 4}{4}} \sqrt[4]{x}}{4}$

Combine the numerators over the common denominator.

$\frac{x^{3} y^{\frac{5 + 2 \cdot 4}{4}} \sqrt[4]{x}}{4}$

Simplify the numerator.

Multiply $2$ by $4$ .

$\frac{x^{3} y^{\frac{5 + 8}{4}} \sqrt[4]{x}}{4}$

Add $5$ and $8$ .

$\frac{x^{3} y^{\frac{13}{4}} \sqrt[4]{x}}{4}$