Algebra Examples

$\frac{\sqrt{2}}{\sqrt{40}}$

Step 1

Combine $\sqrt{2}$ and $\sqrt{40}$ into a single radical.

$\sqrt{\frac{2}{40}}$

Step 2

Cancel the common factor of $2$ and $40$ .

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Step 2.1

Factor $2$ out of $2$ .

$\sqrt{\frac{2 (1)}{40}}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $2$ out of $40$ .

$\sqrt{\frac{2 \cdot 1}{2 \cdot 20}}$

Step 2.2.2

Cancel the common factor.

$\sqrt{\frac{2 \cdot 1}{2 \cdot 20}}$

Step 2.2.3

Rewrite the expression.

$\sqrt{\frac{1}{20}}$

Step 3

Rewrite $\sqrt{\frac{1}{20}}$ as $\frac{\sqrt{1}}{\sqrt{20}}$ .

$\frac{\sqrt{1}}{\sqrt{20}}$

Step 4

Any root of $1$ is $1$ .

$\frac{1}{\sqrt{20}}$

Step 5

Simplify the denominator.

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Step 5.1

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 5.1.1

Factor $4$ out of $20$ .

$\frac{1}{\sqrt{4 (5)}}$

Step 5.1.2

Rewrite $4$ as $2^{2}$ .

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

Step 5.2

Pull terms out from under the radical.

$\frac{1}{2 \sqrt{5}}$

Step 6

Multiply $\frac{1}{2 \sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{1}{2 \sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{1}{2 \sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{\sqrt{5}}{2 \sqrt{5} \sqrt{5}}$

Step 7.2

Move $\sqrt{5}$ .

$\frac{\sqrt{5}}{2 (\sqrt{5} \sqrt{5})}$

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

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

$\frac{\sqrt{5}}{2 {\sqrt{5}}^{1 + 1}}$

Step 7.6

Add $1$ and $1$ .

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

Step 7.7

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

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Step 7.7.1

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

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

Step 7.7.2

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

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

Step 7.7.3

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

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

Step 7.7.4

Cancel the common factor of $2$ .

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Step 7.7.4.1

Cancel the common factor.

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

Step 7.7.4.2

Rewrite the expression.

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

Step 7.7.5

Evaluate the exponent.

$\frac{\sqrt{5}}{2 \cdot 5}$

Step 8

Multiply $2$ by $5$ .

$\frac{\sqrt{5}}{10}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{5}}{10}$

Decimal Form:

$0.22360679 \dots$