Simplify eighth root of x^7y^9 square root of xy

$\sqrt[8]{x^{7} y^{9}} \sqrt{x y}$

Step 1

Rewrite $x^{7} y^{9}$ as $y^{8} (x^{7} y)$ .

Factor out $y^{8}$ .

$\sqrt[8]{x^{7} (y^{8} y)} \sqrt{x y}$

Reorder $x^{7}$ and $y^{8}$ .

$\sqrt[8]{y^{8} x^{7} y} \sqrt{x y}$

Add parentheses.

$\sqrt[8]{y^{8} (x^{7} y)} \sqrt{x y}$

Step 2

Pull terms out from under the radical.

$y \sqrt[8]{x^{7} y} \sqrt{x y}$

Step 3

Multiply $y \sqrt[8]{x^{7} y} \sqrt{x y}$ .

Rewrite the expression using the least common index of $8$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x y}$ as ${(x y)}^{\frac{1}{2}}$ .

$y ({(x y)}^{\frac{1}{2}} \sqrt[8]{x^{7} y})$

Rewrite ${(x y)}^{\frac{1}{2}}$ as ${(x y)}^{\frac{4}{8}}$ .

$y ({(x y)}^{\frac{4}{8}} \sqrt[8]{x^{7} y})$

Rewrite ${(x y)}^{\frac{4}{8}}$ as $\sqrt[8]{{(x y)}^{4}}$ .

$y (\sqrt[8]{{(x y)}^{4}} \sqrt[8]{x^{7} y})$

Combine using the product rule for radicals.

$y \sqrt[8]{{(x y)}^{4} (x^{7} y)}$

Apply the product rule to $x y$ .

$y \sqrt[8]{x^{4} y^{4} x^{7} y}$

Combine exponents.

Multiply $x^{4}$ by $x^{7}$ by adding the exponents.

Move $x^{7}$ .

$y \sqrt[8]{x^{7} x^{4} y^{4} y}$

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

$y \sqrt[8]{x^{7 + 4} y^{4} y}$

Add $7$ and $4$ .

$y \sqrt[8]{x^{11} y^{4} y}$

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

$y \sqrt[8]{x^{11} (y^{1} y^{4})}$

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

$y \sqrt[8]{x^{11} y^{1 + 4}}$

Add $1$ and $4$ .

$y \sqrt[8]{x^{11} y^{5}}$

Step 4

Rewrite $x^{11} y^{5}$ as $x^{8} (x^{3} y^{5})$ .

Factor out $x^{8}$ .

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

Add parentheses.

$y \sqrt[8]{x^{8} (x^{3} y^{5})}$

Step 5

Pull terms out from under the radical.

$y (x \sqrt[8]{x^{3} y^{5}})$

Step 6

Reorder factors in $y (x \sqrt[8]{x^{3} y^{5}})$ .

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

Simplify eighth root of x^7y^9 square root of xy

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