Trigonometry Examples

$\frac{\sqrt{108 x^{5} y^{8}}}{\sqrt{6 x y^{5}}}$

Step 1

Combine $\sqrt{108 x^{5} y^{8}}$ and $\sqrt{6 x y^{5}}$ into a single radical.

$\sqrt{\frac{108 x^{5} y^{8}}{6 x y^{5}}}$

Step 2

Reduce the expression $\frac{108 x^{5} y^{8}}{6 x y^{5}}$ by cancelling the common factors.

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Step 2.1

Factor $6$ out of $108 x^{5} y^{8}$ .

$\sqrt{\frac{6 (18 x^{5} y^{8})}{6 x y^{5}}}$

Step 2.2

Factor $6$ out of $6 x y^{5}$ .

$\sqrt{\frac{6 (18 x^{5} y^{8})}{6 (x y^{5})}}$

Step 2.3

Cancel the common factor.

$\sqrt{\frac{6 (18 x^{5} y^{8})}{6 (x y^{5})}}$

Step 2.4

Rewrite the expression.

$\sqrt{\frac{18 x^{5} y^{8}}{x y^{5}}}$

Step 3

Cancel the common factor of $x^{5}$ and $x$ .

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Step 3.1

Factor $x$ out of $18 x^{5} y^{8}$ .

$\sqrt{\frac{x (18 x^{4} y^{8})}{x y^{5}}}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $x$ out of $x y^{5}$ .

$\sqrt{\frac{x (18 x^{4} y^{8})}{x (y^{5})}}$

Step 3.2.2

Cancel the common factor.

$\sqrt{\frac{x (18 x^{4} y^{8})}{x y^{5}}}$

Step 3.2.3

Rewrite the expression.

$\sqrt{\frac{18 x^{4} y^{8}}{y^{5}}}$

Step 4

Cancel the common factor of $y^{8}$ and $y^{5}$ .

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Step 4.1

Factor $y^{5}$ out of $18 x^{4} y^{8}$ .

$\sqrt{\frac{y^{5} (18 x^{4} y^{3})}{y^{5}}}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Multiply by $1$ .

$\sqrt{\frac{y^{5} (18 x^{4} y^{3})}{y^{5} \cdot 1}}$

Step 4.2.2

Cancel the common factor.

$\sqrt{\frac{y^{5} (18 x^{4} y^{3})}{y^{5} \cdot 1}}$

Step 4.2.3

Rewrite the expression.

$\sqrt{\frac{18 x^{4} y^{3}}{1}}$

Step 4.2.4

Divide $18 x^{4} y^{3}$ by $1$ .

$\sqrt{18 x^{4} y^{3}}$

Step 5

Rewrite $18 x^{4} y^{3}$ as ${(3 x^{2} y)}^{2} \cdot (2 y)$ .

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Step 5.1

Factor $9$ out of $18$ .

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

Step 5.2

Rewrite $9$ as $3^{2}$ .

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

Step 5.3

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

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

Step 5.4

Factor out $y^{2}$ .

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

Step 5.5

Move $2$ .

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

Step 5.6

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

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

Step 5.7

Add parentheses.

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

Step 6

Pull terms out from under the radical.

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