Basic Math Examples

$\sqrt{\frac{5}{100}} + \sqrt{\frac{5}{4}}$

Step 1

Simplify each term.

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Step 1.1

Cancel the common factor of $5$ and $100$ .

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Step 1.1.1

Factor $5$ out of $5$ .

$\sqrt{\frac{5 (1)}{100}} + \sqrt{\frac{5}{4}}$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Factor $5$ out of $100$ .

$\sqrt{\frac{5 \cdot 1}{5 \cdot 20}} + \sqrt{\frac{5}{4}}$

Step 1.1.2.2

Cancel the common factor.

$\sqrt{\frac{5 \cdot 1}{5 \cdot 20}} + \sqrt{\frac{5}{4}}$

Step 1.1.2.3

Rewrite the expression.

$\sqrt{\frac{1}{20}} + \sqrt{\frac{5}{4}}$

Step 1.2

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

$\frac{\sqrt{1}}{\sqrt{20}} + \sqrt{\frac{5}{4}}$

Step 1.3

Any root of $1$ is $1$ .

$\frac{1}{\sqrt{20}} + \sqrt{\frac{5}{4}}$

Step 1.4

Simplify the denominator.

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Step 1.4.1

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

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Step 1.4.1.1

Factor $4$ out of $20$ .

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

Step 1.4.1.2

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

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

Step 1.4.2

Pull terms out from under the radical.

$\frac{1}{2 \sqrt{5}} + \sqrt{\frac{5}{4}}$

Step 1.5

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

$\frac{1}{2 \sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} + \sqrt{\frac{5}{4}}$

Step 1.6

Combine and simplify the denominator.

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Step 1.6.1

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

$\frac{\sqrt{5}}{2 \sqrt{5} \sqrt{5}} + \sqrt{\frac{5}{4}}$

Step 1.6.2

Move $\sqrt{5}$ .

$\frac{\sqrt{5}}{2 (\sqrt{5} \sqrt{5})} + \sqrt{\frac{5}{4}}$

Step 1.6.3

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

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

Step 1.6.4

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

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

Step 1.6.5

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

$\frac{\sqrt{5}}{2 {\sqrt{5}}^{1 + 1}} + \sqrt{\frac{5}{4}}$

Step 1.6.6

Add $1$ and $1$ .

$\frac{\sqrt{5}}{2 {\sqrt{5}}^{2}} + \sqrt{\frac{5}{4}}$

Step 1.6.7

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

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Step 1.6.7.1

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

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

Step 1.6.7.2

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

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

Step 1.6.7.3

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

$\frac{\sqrt{5}}{2 \cdot 5^{\frac{2}{2}}} + \sqrt{\frac{5}{4}}$

Step 1.6.7.4

Cancel the common factor of $2$ .

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Step 1.6.7.4.1

Cancel the common factor.

$\frac{\sqrt{5}}{2 \cdot 5^{\frac{2}{2}}} + \sqrt{\frac{5}{4}}$

Step 1.6.7.4.2

Rewrite the expression.

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

Step 1.6.7.5

Evaluate the exponent.

$\frac{\sqrt{5}}{2 \cdot 5} + \sqrt{\frac{5}{4}}$

Step 1.7

Multiply $2$ by $5$ .

$\frac{\sqrt{5}}{10} + \sqrt{\frac{5}{4}}$

Step 1.8

Rewrite $\sqrt{\frac{5}{4}}$ as $\frac{\sqrt{5}}{\sqrt{4}}$ .

$\frac{\sqrt{5}}{10} + \frac{\sqrt{5}}{\sqrt{4}}$

Step 1.9

Simplify the denominator.

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Step 1.9.1

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

$\frac{\sqrt{5}}{10} + \frac{\sqrt{5}}{\sqrt{2^{2}}}$

Step 1.9.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{\sqrt{5}}{10} + \frac{\sqrt{5}}{2}$

Step 2

To write $\frac{\sqrt{5}}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{\sqrt{5}}{10} + \frac{\sqrt{5}}{2} \cdot \frac{5}{5}$

Step 3

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

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

$\frac{\sqrt{5}}{10} + \frac{\sqrt{5} \cdot 5}{2 \cdot 5}$

Step 3.2

Multiply $2$ by $5$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Move $5$ to the left of $\sqrt{5}$ .

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

Step 5.2

Add $\sqrt{5}$ and $5 \sqrt{5}$ .

$\frac{6 \sqrt{5}}{10}$

Step 6

Cancel the common factor of $6$ and $10$ .

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Step 6.1

Factor $2$ out of $6 \sqrt{5}$ .

$\frac{2 (3 \sqrt{5})}{10}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $2$ out of $10$ .

$\frac{2 (3 \sqrt{5})}{2 (5)}$

Step 6.2.2

Cancel the common factor.

$\frac{2 (3 \sqrt{5})}{2 \cdot 5}$

Step 6.2.3

Rewrite the expression.

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.34164078 \dots$