Simplify ( fourth root of 9x^2y^6)( fourth root of 9x^6y^2)

$(\sqrt[4]{9 x^{2} y^{6}}) (\sqrt[4]{9 x^{6} y^{2}})$

Step 1

Rewrite $9 x^{2} y^{6}$ as $y^{4} \cdot (9 x^{2} y^{2})$ .

Factor out $y^{4}$ .

$\sqrt[4]{9 x^{2} (y^{4} y^{2})} \sqrt[4]{9 x^{6} y^{2}}$

Move $x^{2}$ .

$\sqrt[4]{9 y^{4} x^{2} y^{2}} \sqrt[4]{9 x^{6} y^{2}}$

Reorder $9$ and $y^{4}$ .

$\sqrt[4]{y^{4} \cdot 9 x^{2} y^{2}} \sqrt[4]{9 x^{6} y^{2}}$

Add parentheses.

$\sqrt[4]{y^{4} \cdot 9 (x^{2} y^{2})} \sqrt[4]{9 x^{6} y^{2}}$

Add parentheses.

$\sqrt[4]{y^{4} \cdot (9 x^{2} y^{2})} \sqrt[4]{9 x^{6} y^{2}}$

Step 2

Pull terms out from under the radical.

$y \sqrt[4]{9 x^{2} y^{2}} \sqrt[4]{9 x^{6} y^{2}}$

Step 3

Rewrite $9 x^{2} y^{2}$ as ${(3 x y)}^{2}$ .

$y \sqrt[4]{{(3 x y)}^{2}} \sqrt[4]{9 x^{6} y^{2}}$

Step 4

Rewrite $\sqrt[4]{{(3 x y)}^{2}}$ as $\sqrt{\sqrt{{(3 x y)}^{2}}}$ .

$y \sqrt{\sqrt{{(3 x y)}^{2}}} \sqrt[4]{9 x^{6} y^{2}}$

Step 5

Pull terms out from under the radical, assuming positive real numbers.

$y \sqrt{3 x y} \sqrt[4]{9 x^{6} y^{2}}$

Step 6

Rewrite $9 x^{6} y^{2}$ as $x^{4} \cdot (9 x^{2} y^{2})$ .

Factor out $x^{4}$ .

$y \sqrt{3 x y} \sqrt[4]{9 (x^{4} x^{2}) y^{2}}$

Reorder $9$ and $x^{4}$ .

$y \sqrt{3 x y} \sqrt[4]{x^{4} \cdot 9 x^{2} y^{2}}$

Add parentheses.

$y \sqrt{3 x y} \sqrt[4]{x^{4} \cdot 9 (x^{2} y^{2})}$

Add parentheses.

$y \sqrt{3 x y} \sqrt[4]{x^{4} \cdot (9 x^{2} y^{2})}$

Step 7

Pull terms out from under the radical.

$y \sqrt{3 x y} (x \sqrt[4]{9 x^{2} y^{2}})$

Step 8

Rewrite $9 x^{2} y^{2}$ as ${(3 x y)}^{2}$ .

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

Step 9

Rewrite $\sqrt[4]{{(3 x y)}^{2}}$ as $\sqrt{\sqrt{{(3 x y)}^{2}}}$ .

$y \sqrt{3 x y} (x \sqrt{\sqrt{{(3 x y)}^{2}}})$

Step 10

Pull terms out from under the radical, assuming positive real numbers.

$y \sqrt{3 x y} (x \sqrt{3 x y})$

Step 11

Multiply $y \sqrt{3 x y} (x \sqrt{3 x y})$ .

Raise $\sqrt{3 x y}$ to the power of $1$ .

$y (x ({\sqrt{3 x y}}^{1} \sqrt{3 x y}))$

Raise $\sqrt{3 x y}$ to the power of $1$ .

$y (x ({\sqrt{3 x y}}^{1} {\sqrt{3 x y}}^{1}))$

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

$y (x {\sqrt{3 x y}}^{1 + 1})$

Add $1$ and $1$ .

$y (x {\sqrt{3 x y}}^{2})$

Step 12

Rewrite ${\sqrt{3 x y}}^{2}$ as $3 x y$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x y}$ as ${(3 x y)}^{\frac{1}{2}}$ .

$y (x {({(3 x y)}^{\frac{1}{2}})}^{2})$

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

$y (x {(3 x y)}^{\frac{1}{2} \cdot 2})$

Combine $\frac{1}{2}$ and $2$ .

$y (x {(3 x y)}^{\frac{2}{2}})$

Cancel the common factor of $2$ .

Cancel the common factor.

$y (x {(3 x y)}^{\frac{2}{2}})$

Rewrite the expression.

$y (x {(3 x y)}^{1})$

Simplify.

$y (x (3 x y))$

Step 13

Rewrite using the commutative property of multiplication.

$3 y (x \cdot x y)$

Step 14

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

$3 (y \cdot y) (x \cdot x)$

Multiply $y$ by $y$ .

$3 y^{2} (x \cdot x)$

Step 15

Multiply $x$ by $x$ .

$3 y^{2} x^{2}$

Simplify ( fourth root of 9x^2y^6)( fourth root of 9x^6y^2)

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