Basic Math Examples

$\sqrt{\frac{1}{20}} + \sqrt{\frac{5}{5}}$

Step 1

Simplify each term.

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Step 1.1

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

$\frac{\sqrt{1}}{\sqrt{20}} + \sqrt{\frac{5}{5}}$

Step 1.2

Any root of $1$ is $1$ .

$\frac{1}{\sqrt{20}} + \sqrt{\frac{5}{5}}$

Step 1.3

Simplify the denominator.

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Step 1.3.1

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

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Step 1.3.1.1

Factor $4$ out of $20$ .

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

Step 1.3.1.2

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

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

Step 1.3.2

Pull terms out from under the radical.

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

Step 1.4

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

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

Step 1.5

Combine and simplify the denominator.

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Step 1.5.1

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

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

Step 1.5.2

Move $\sqrt{5}$ .

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

Step 1.5.3

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

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

Step 1.5.4

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

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

Step 1.5.5

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

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

Step 1.5.6

Add $1$ and $1$ .

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

Step 1.5.7

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

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Step 1.5.7.1

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

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

Step 1.5.7.2

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

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

Step 1.5.7.3

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

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

Step 1.5.7.4

Cancel the common factor of $2$ .

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Step 1.5.7.4.1

Cancel the common factor.

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

Step 1.5.7.4.2

Rewrite the expression.

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

Step 1.5.7.5

Evaluate the exponent.

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

Step 1.6

Multiply $2$ by $5$ .

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

Step 1.7

Divide $5$ by $5$ .

$\frac{\sqrt{5}}{10} + \sqrt{1}$

Step 1.8

Any root of $1$ is $1$ .

$\frac{\sqrt{5}}{10} + 1$

Step 2

Simplify the expression.

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Step 2.1

Write $1$ as a fraction with a common denominator.

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

Step 2.2

Combine the numerators over the common denominator.

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.22360679 \dots$