Algebra Examples

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

Step 1

Rewrite $\frac{3}{y^{5}}$ as ${(\frac{1}{y^{2}})}^{2} \frac{3}{y}$ .

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Step 1.1

Factor the perfect power $1^{2}$ out of $3$ .

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

Step 1.2

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

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

Step 1.3

Rearrange the fraction $\frac{1^{2} \cdot 3}{{(y^{2})}^{2} y}$ .

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

Step 2

Pull terms out from under the radical.

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

Step 3

Remove non-negative terms from the absolute value.

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

Step 4

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

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

Step 5

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

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

Step 6

Combine.

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

Step 7

Multiply $\sqrt{3}$ by $1$ .

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

Step 8

Multiply $\frac{\sqrt{3}}{y^{2} \sqrt{y}}$ by $\frac{\sqrt{y}}{\sqrt{y}}$ .

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

Step 9

Combine and simplify the denominator.

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Step 9.1

Multiply $\frac{\sqrt{3}}{y^{2} \sqrt{y}}$ by $\frac{\sqrt{y}}{\sqrt{y}}$ .

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

Step 9.2

Move $\sqrt{y}$ .

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

Step 9.3

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

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

Step 9.4

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

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

Step 9.5

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

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

Step 9.6

Add $1$ and $1$ .

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

Step 9.7

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

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Step 9.7.1

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

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

Step 9.7.2

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

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

Step 9.7.3

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

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

Step 9.7.4

Cancel the common factor of $2$ .

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Step 9.7.4.1

Cancel the common factor.

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

Step 9.7.4.2

Rewrite the expression.

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

Step 9.7.5

Simplify.

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

Step 10

Multiply $y^{2}$ by $y$ by adding the exponents.

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Step 10.1

Multiply $y^{2}$ by $y$ .

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Step 10.1.1

Raise $y$ to the power of $1$ .

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

Step 10.1.2

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

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

Step 10.2

Add $2$ and $1$ .

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

Step 11

Combine using the product rule for radicals.

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