Algebra Examples

$\sqrt{\frac{80 x^{3}}{25 y^{2}}}$

Step 1

Reduce the expression $\frac{80 x^{3}}{25 y^{2}}$ by cancelling the common factors.

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Step 1.1

Factor $5$ out of $80 x^{3}$ .

$\sqrt{\frac{5 (16 x^{3})}{25 y^{2}}}$

Step 1.2

Factor $5$ out of $25 y^{2}$ .

$\sqrt{\frac{5 (16 x^{3})}{5 (5 y^{2})}}$

Step 1.3

Cancel the common factor.

$\sqrt{\frac{5 (16 x^{3})}{5 (5 y^{2})}}$

Step 1.4

Rewrite the expression.

$\sqrt{\frac{16 x^{3}}{5 y^{2}}}$

Step 2

Rewrite $\frac{16 x^{3}}{5 y^{2}}$ as ${(\frac{4 x}{y})}^{2} \frac{x}{5}$ .

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Step 2.1

Factor the perfect power ${(4 x)}^{2}$ out of $16 x^{3}$ .

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

Step 2.2

Factor the perfect power $y^{2}$ out of $5 y^{2}$ .

$\sqrt{\frac{{(4 x)}^{2} x}{y^{2} \cdot 5}}$

Step 2.3

Rearrange the fraction $\frac{{(4 x)}^{2} x}{y^{2} \cdot 5}$ .

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

Step 3

Pull terms out from under the radical.

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

Step 4

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

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

Step 5

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

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

Step 6

Combine and simplify the denominator.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

Add $1$ and $1$ .

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

Step 6.6

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

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Step 6.6.1

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

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

Step 6.6.2

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

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

Step 6.6.3

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

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

Step 6.6.4

Cancel the common factor of $2$ .

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Step 6.6.4.1

Cancel the common factor.

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

Step 6.6.4.2

Rewrite the expression.

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

Step 6.6.5

Evaluate the exponent.

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

Step 7

Combine using the product rule for radicals.

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

Step 8

Multiply $\frac{4 x}{y}$ by $\frac{\sqrt{x \cdot 5}}{5}$ .

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

Step 9

Reorder.

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Step 9.1

Move $5$ to the left of $x$ .

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

Step 9.2

Move $5$ to the left of $y$ .

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