Algebra Examples

$\frac{6 \sqrt{45 y^{3}}}{3 \sqrt{5 x}}$

Step 1

Cancel the common factor of $6$ and $3$ .

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Step 1.1

Factor $3$ out of $6 \sqrt{45 y^{3}}$ .

$\frac{3 (2 \sqrt{45 y^{3}})}{3 \sqrt{5 x}}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $3$ out of $3 \sqrt{5 x}$ .

$\frac{3 (2 \sqrt{45 y^{3}})}{3 (\sqrt{5 x})}$

Step 1.2.2

Cancel the common factor.

$\frac{3 (2 \sqrt{45 y^{3}})}{3 \sqrt{5 x}}$

Step 1.2.3

Rewrite the expression.

$\frac{2 \sqrt{45 y^{3}}}{\sqrt{5 x}}$

Step 2

Combine $\sqrt{45 y^{3}}$ and $\sqrt{5 x}$ into a single radical.

$2 \sqrt{\frac{45 y^{3}}{5 x}}$

Step 3

Reduce the expression $\frac{45 y^{3}}{5 x}$ by cancelling the common factors.

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Step 3.1

Factor $5$ out of $45 y^{3}$ .

$2 \sqrt{\frac{5 (9 y^{3})}{5 x}}$

Step 3.2

Factor $5$ out of $5 x$ .

$2 \sqrt{\frac{5 (9 y^{3})}{5 (x)}}$

Step 3.3

Cancel the common factor.

$2 \sqrt{\frac{5 (9 y^{3})}{5 x}}$

Step 3.4

Rewrite the expression.

$2 \sqrt{\frac{9 y^{3}}{x}}$

Step 4

Rewrite $\sqrt{\frac{9 y^{3}}{x}}$ as $\frac{\sqrt{9 y^{3}}}{\sqrt{x}}$ .

$2 \frac{\sqrt{9 y^{3}}}{\sqrt{x}}$

Step 5

Simplify the numerator.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

Factor out $y^{2}$ .

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

Step 5.1.3

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

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

Step 5.2

Pull terms out from under the radical.

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

Step 6

Multiply $\frac{3 y \sqrt{y}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

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

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{3 y \sqrt{y}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

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

Step 7.2

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

$2 \frac{3 y \sqrt{y} \sqrt{x}}{{\sqrt{x}}^{1} \sqrt{x}}$

Step 7.3

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

$2 \frac{3 y \sqrt{y} \sqrt{x}}{{\sqrt{x}}^{1} {\sqrt{x}}^{1}}$

Step 7.4

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

$2 \frac{3 y \sqrt{y} \sqrt{x}}{{\sqrt{x}}^{1 + 1}}$

Step 7.5

Add $1$ and $1$ .

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

Step 7.6

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

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Step 7.6.1

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

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

Step 7.6.2

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

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

Step 7.6.3

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

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

Step 7.6.4

Cancel the common factor of $2$ .

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Step 7.6.4.1

Cancel the common factor.

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

Step 7.6.4.2

Rewrite the expression.

$2 \frac{3 y \sqrt{y} \sqrt{x}}{x^{1}}$

Step 7.6.5

Simplify.

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

Step 8

Combine using the product rule for radicals.

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

Step 9

Multiply $2 \frac{3 y \sqrt{x y}}{x}$ .

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Step 9.1

Combine $2$ and $\frac{3 y \sqrt{x y}}{x}$ .

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

Step 9.2

Multiply $3$ by $2$ .

$\frac{6 (y \sqrt{x y})}{x}$

$\frac{6 y \sqrt{x y}}{x}$