Algebra Examples

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

Step 1

Combine $\sqrt{3 x y^{2}}$ and $\sqrt{5 x y^{3}}$ into a single radical.

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

Step 2

Reduce the expression $\frac{3 x y^{2}}{5 x y^{3}}$ by cancelling the common factors.

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Step 2.1

Cancel the common factor.

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

Step 2.2

Rewrite the expression.

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

Step 3

Cancel the common factor of $y^{2}$ and $y^{3}$ .

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Step 3.1

Factor $y^{2}$ out of $3 y^{2}$ .

$\sqrt{\frac{y^{2} \cdot 3}{5 y^{3}}}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $y^{2}$ out of $5 y^{3}$ .

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

$\sqrt{\frac{3}{5 y}}$

Step 4

Rewrite $\sqrt{\frac{3}{5 y}}$ as $\frac{\sqrt{3}}{\sqrt{5 y}}$ .

$\frac{\sqrt{3}}{\sqrt{5 y}}$

Step 5

Multiply $\frac{\sqrt{3}}{\sqrt{5 y}}$ by $\frac{\sqrt{5 y}}{\sqrt{5 y}}$ .

$\frac{\sqrt{3}}{\sqrt{5 y}} \cdot \frac{\sqrt{5 y}}{\sqrt{5 y}}$

Step 6

Combine and simplify the denominator.

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Step 6.1

Multiply $\frac{\sqrt{3}}{\sqrt{5 y}}$ by $\frac{\sqrt{5 y}}{\sqrt{5 y}}$ .

$\frac{\sqrt{3} \sqrt{5 y}}{\sqrt{5 y} \sqrt{5 y}}$

Step 6.2

Raise $\sqrt{5 y}$ to the power of $1$ .

$\frac{\sqrt{3} \sqrt{5 y}}{{\sqrt{5 y}}^{1} \sqrt{5 y}}$

Step 6.3

Raise $\sqrt{5 y}$ to the power of $1$ .

$\frac{\sqrt{3} \sqrt{5 y}}{{\sqrt{5 y}}^{1} {\sqrt{5 y}}^{1}}$

Step 6.4

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

$\frac{\sqrt{3} \sqrt{5 y}}{{\sqrt{5 y}}^{1 + 1}}$

Step 6.5

Add $1$ and $1$ .

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

Step 6.6

Rewrite ${\sqrt{5 y}}^{2}$ as $5 y$ .

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Step 6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 y}$ as ${(5 y)}^{\frac{1}{2}}$ .

$\frac{\sqrt{3} \sqrt{5 y}}{{({(5 y)}^{\frac{1}{2}})}^{2}}$

Step 6.6.2

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

$\frac{\sqrt{3} \sqrt{5 y}}{{(5 y)}^{\frac{1}{2} \cdot 2}}$

Step 6.6.3

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

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

Step 6.6.4

Cancel the common factor of $2$ .

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Step 6.6.4.1

Cancel the common factor.

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

Step 6.6.4.2

Rewrite the expression.

$\frac{\sqrt{3} \sqrt{5 y}}{{(5 y)}^{1}}$

Step 6.6.5

Simplify.

$\frac{\sqrt{3} \sqrt{5 y}}{5 y}$

Step 7

Simplify the numerator.

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Step 7.1

Combine using the product rule for radicals.

$\frac{\sqrt{3 (5 y)}}{5 y}$

Step 7.2

Multiply $3$ by $5$ .

$\frac{\sqrt{15 y}}{5 y}$