Algebra Examples

$\frac{\sqrt{32 x^{9} y^{2}}}{\sqrt{2 x y^{- 2}}}$

Step 1

Combine $\sqrt{32 x^{9} y^{2}}$ and $\sqrt{2 x y^{- 2}}$ into a single radical.

$\sqrt{\frac{32 x^{9} y^{2}}{2 x y^{- 2}}}$

Step 2

Reduce the expression $\frac{32 x^{9} y^{2}}{2 x y^{- 2}}$ by cancelling the common factors.

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Step 2.1

Factor $2$ out of $32 x^{9} y^{2}$ .

$\sqrt{\frac{2 (16 x^{9} y^{2})}{2 x y^{- 2}}}$

Step 2.2

Factor $2$ out of $2 x y^{- 2}$ .

$\sqrt{\frac{2 (16 x^{9} y^{2})}{2 (x y^{- 2})}}$

Step 2.3

Cancel the common factor.

$\sqrt{\frac{2 (16 x^{9} y^{2})}{2 (x y^{- 2})}}$

Step 2.4

Rewrite the expression.

$\sqrt{\frac{16 x^{9} y^{2}}{x y^{- 2}}}$

Step 3

Move $y^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\sqrt{\frac{16 x^{9} y^{2} y^{2}}{x}}$

Step 4

Multiply $y^{2}$ by $y^{2}$ by adding the exponents.

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Step 4.1

Move $y^{2}$ .

$\sqrt{\frac{16 x^{9} (y^{2} y^{2})}{x}}$

Step 4.2

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

$\sqrt{\frac{16 x^{9} y^{2 + 2}}{x}}$

Step 4.3

Add $2$ and $2$ .

$\sqrt{\frac{16 x^{9} y^{4}}{x}}$

Step 5

Cancel the common factor of $x^{9}$ and $x$ .

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Step 5.1

Factor $x$ out of $16 x^{9} y^{4}$ .

$\sqrt{\frac{x (16 x^{8} y^{4})}{x}}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

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

$\sqrt{\frac{x (16 x^{8} y^{4})}{x^{1}}}$

Step 5.2.2

Factor $x$ out of $x^{1}$ .

$\sqrt{\frac{x (16 x^{8} y^{4})}{x \cdot 1}}$

Step 5.2.3

Cancel the common factor.

$\sqrt{\frac{x (16 x^{8} y^{4})}{x \cdot 1}}$

Step 5.2.4

Rewrite the expression.

$\sqrt{\frac{16 x^{8} y^{4}}{1}}$

Step 5.2.5

Divide $16 x^{8} y^{4}$ by $1$ .

$\sqrt{16 x^{8} y^{4}}$

Step 6

Rewrite $16 x^{8} y^{4}$ as ${(4 x^{4} y^{2})}^{2}$ .

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

Step 7

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

$4 x^{4} y^{2}$