Trigonometry 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

Factor $2$ out of $2 x y - 2$ .

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Step 2.1

Factor $2$ out of $2 x y$ .

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

Step 2.2

Factor $2$ out of $- 2$ .

$\sqrt{\frac{32 x^{9} y^{2}}{2 (x y) + 2 (- 1)}}$

Step 2.3

Factor $2$ out of $2 (x y) + 2 (- 1)$ .

$\sqrt{\frac{32 x^{9} y^{2}}{2 (x y - 1)}}$

Step 3

Reduce the expression $\frac{32 x^{9} y^{2}}{2 (x y - 1)}$ by cancelling the common factors.

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Step 3.1

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

$\sqrt{\frac{2 (16 x^{9} y^{2})}{2 (x y - 1)}}$

Step 3.2

Cancel the common factor.

$\sqrt{\frac{2 (16 x^{9} y^{2})}{2 (x y - 1)}}$

Step 3.3

Rewrite the expression.

$\sqrt{\frac{16 x^{9} y^{2}}{x y - 1}}$

Step 4

Rewrite $\sqrt{\frac{16 x^{9} y^{2}}{x y - 1}}$ as $\frac{\sqrt{16 x^{9} y^{2}}}{\sqrt{x y - 1}}$ .

$\frac{\sqrt{16 x^{9} y^{2}}}{\sqrt{x y - 1}}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite $16 x^{9} y^{2}$ as ${(4 x^{4} y)}^{2} x$ .

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Step 5.1.1

Rewrite $16$ as $4^{2}$ .

$\frac{\sqrt{4^{2} x^{9} y^{2}}}{\sqrt{x y - 1}}$

Step 5.1.2

Factor out $x^{8}$ .

$\frac{\sqrt{4^{2} (x^{8} x) y^{2}}}{\sqrt{x y - 1}}$

Step 5.1.3

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

$\frac{\sqrt{4^{2} ({(x^{4})}^{2} x) y^{2}}}{\sqrt{x y - 1}}$

Step 5.1.4

Move $x$ .

$\frac{\sqrt{4^{2} ({(x^{4})}^{2}) y^{2} x}}{\sqrt{x y - 1}}$

Step 5.1.5

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

$\frac{\sqrt{{(4 x^{4} y)}^{2} x}}{\sqrt{x y - 1}}$

Step 5.2

Pull terms out from under the radical.

$\frac{4 x^{4} y \sqrt{x}}{\sqrt{x y - 1}}$

Step 6

Multiply $\frac{4 x^{4} y \sqrt{x}}{\sqrt{x y - 1}}$ by $\frac{\sqrt{x y - 1}}{\sqrt{x y - 1}}$ .

$\frac{4 x^{4} y \sqrt{x}}{\sqrt{x y - 1}} \cdot \frac{\sqrt{x y - 1}}{\sqrt{x y - 1}}$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{4 x^{4} y \sqrt{x}}{\sqrt{x y - 1}}$ by $\frac{\sqrt{x y - 1}}{\sqrt{x y - 1}}$ .

$\frac{4 x^{4} y \sqrt{x} \sqrt{x y - 1}}{\sqrt{x y - 1} \sqrt{x y - 1}}$

Step 7.2

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

$\frac{4 x^{4} y \sqrt{x} \sqrt{x y - 1}}{{\sqrt{x y - 1}}^{1} \sqrt{x y - 1}}$

Step 7.3

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

$\frac{4 x^{4} y \sqrt{x} \sqrt{x y - 1}}{{\sqrt{x y - 1}}^{1} {\sqrt{x y - 1}}^{1}}$

Step 7.4

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

$\frac{4 x^{4} y \sqrt{x} \sqrt{x y - 1}}{{\sqrt{x y - 1}}^{1 + 1}}$

Step 7.5

Add $1$ and $1$ .

$\frac{4 x^{4} y \sqrt{x} \sqrt{x y - 1}}{{\sqrt{x y - 1}}^{2}}$

Step 7.6

Rewrite ${\sqrt{x y - 1}}^{2}$ as $x y - 1$ .

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Step 7.6.1

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

$\frac{4 x^{4} y \sqrt{x} \sqrt{x y - 1}}{{({(x y - 1)}^{\frac{1}{2}})}^{2}}$

Step 7.6.2

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

$\frac{4 x^{4} y \sqrt{x} \sqrt{x y - 1}}{{(x y - 1)}^{\frac{1}{2} \cdot 2}}$

Step 7.6.3

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

$\frac{4 x^{4} y \sqrt{x} \sqrt{x y - 1}}{{(x y - 1)}^{\frac{2}{2}}}$

Step 7.6.4

Cancel the common factor of $2$ .

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Step 7.6.4.1

Cancel the common factor.

$\frac{4 x^{4} y \sqrt{x} \sqrt{x y - 1}}{{(x y - 1)}^{\frac{2}{2}}}$

Step 7.6.4.2

Rewrite the expression.

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

Step 7.6.5

Simplify.

$\frac{4 x^{4} y \sqrt{x} \sqrt{x y - 1}}{x y - 1}$

Step 8

Combine using the product rule for radicals.

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

Step 9

Reorder factors in $\frac{4 x^{4} y \sqrt{(x y - 1) x}}{x y - 1}$ .

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