Algebra Examples

$\frac{\sqrt{3 x^{2} y^{3}}}{4 \sqrt{5 x y^{3}}}$

Step 1

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

$\frac{\sqrt{\frac{3 x^{2} y^{3}}{5 x y^{3}}}}{4}$

Step 2

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

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Step 2.1

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

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

Step 2.2

Cancel the common factors.

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Step 2.2.1

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

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

Step 2.2.2

Cancel the common factor.

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

Step 2.2.3

Rewrite the expression.

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

Step 3

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

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Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

$\frac{\sqrt{\frac{3 x}{5}}}{4}$

Step 4

Simplify the numerator.

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Step 4.1

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

$\frac{\frac{\sqrt{3 x}}{\sqrt{5}}}{4}$

Step 4.2

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

$\frac{\frac{\sqrt{3 x}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}}{4}$

Step 4.3

Combine and simplify the denominator.

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Step 4.3.1

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

$\frac{\frac{\sqrt{3 x} \sqrt{5}}{\sqrt{5} \sqrt{5}}}{4}$

Step 4.3.2

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

$\frac{\frac{\sqrt{3 x} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}}{4}$

Step 4.3.3

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

$\frac{\frac{\sqrt{3 x} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}}{4}$

Step 4.3.4

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

$\frac{\frac{\sqrt{3 x} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}}{4}$

Step 4.3.5

Add $1$ and $1$ .

$\frac{\frac{\sqrt{3 x} \sqrt{5}}{{\sqrt{5}}^{2}}}{4}$

Step 4.3.6

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

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Step 4.3.6.1

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

$\frac{\frac{\sqrt{3 x} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}}{4}$

Step 4.3.6.2

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

$\frac{\frac{\sqrt{3 x} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}}{4}$

Step 4.3.6.3

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

$\frac{\frac{\sqrt{3 x} \sqrt{5}}{5^{\frac{2}{2}}}}{4}$

Step 4.3.6.4

Cancel the common factor of $2$ .

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Step 4.3.6.4.1

Cancel the common factor.

$\frac{\frac{\sqrt{3 x} \sqrt{5}}{5^{\frac{2}{2}}}}{4}$

Step 4.3.6.4.2

Rewrite the expression.

$\frac{\frac{\sqrt{3 x} \sqrt{5}}{5^{1}}}{4}$

Step 4.3.6.5

Evaluate the exponent.

$\frac{\frac{\sqrt{3 x} \sqrt{5}}{5}}{4}$

Step 4.4

Simplify the numerator.

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Step 4.4.1

Combine using the product rule for radicals.

$\frac{\frac{\sqrt{3 x \cdot 5}}{5}}{4}$

Step 4.4.2

Multiply $5$ by $3$ .

$\frac{\frac{\sqrt{15 x}}{5}}{4}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sqrt{15 x}}{5} \cdot \frac{1}{4}$

Step 6

Multiply $\frac{\sqrt{15 x}}{5} \cdot \frac{1}{4}$ .

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Step 6.1

Multiply $\frac{\sqrt{15 x}}{5}$ by $\frac{1}{4}$ .

$\frac{\sqrt{15 x}}{5 \cdot 4}$

Step 6.2

Multiply $5$ by $4$ .

$\frac{\sqrt{15 x}}{20}$