Basic Math Examples

$\frac{\sqrt{5} + \sqrt{45}}{3 - \sqrt{20}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $45$ as $3^{2} \cdot 5$ .

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Step 1.1.1

Factor $9$ out of $45$ .

$\frac{\sqrt{5} + \sqrt{9 (5)}}{3 - \sqrt{20}}$

Step 1.1.2

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{5} + \sqrt{3^{2} \cdot 5}}{3 - \sqrt{20}}$

Step 1.2

Pull terms out from under the radical.

$\frac{\sqrt{5} + 3 \sqrt{5}}{3 - \sqrt{20}}$

Step 1.3

Add $\sqrt{5}$ and $3 \sqrt{5}$ .

$\frac{4 \sqrt{5}}{3 - \sqrt{20}}$

Step 2

Simplify the denominator.

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Step 2.1

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

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Step 2.1.1

Factor $4$ out of $20$ .

$\frac{4 \sqrt{5}}{3 - \sqrt{4 (5)}}$

Step 2.1.2

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

$\frac{4 \sqrt{5}}{3 - \sqrt{2^{2} \cdot 5}}$

Step 2.2

Pull terms out from under the radical.

$\frac{4 \sqrt{5}}{3 - (2 \sqrt{5})}$

Step 2.3

Multiply $2$ by $- 1$ .

$\frac{4 \sqrt{5}}{3 - 2 \sqrt{5}}$

Step 3

Multiply $\frac{4 \sqrt{5}}{3 - 2 \sqrt{5}}$ by $\frac{3 + 2 \sqrt{5}}{3 + 2 \sqrt{5}}$ .

$\frac{4 \sqrt{5}}{3 - 2 \sqrt{5}} \cdot \frac{3 + 2 \sqrt{5}}{3 + 2 \sqrt{5}}$

Step 4

Multiply $\frac{4 \sqrt{5}}{3 - 2 \sqrt{5}}$ by $\frac{3 + 2 \sqrt{5}}{3 + 2 \sqrt{5}}$ .

$\frac{4 \sqrt{5} (3 + 2 \sqrt{5})}{(3 - 2 \sqrt{5}) (3 + 2 \sqrt{5})}$

Step 5

Expand the denominator using the FOIL method.

$\frac{4 \sqrt{5} (3 + 2 \sqrt{5})}{9 + 6 \sqrt{5} - 6 \sqrt{5} - 4 {\sqrt{5}}^{2}}$

Step 6

Simplify.

$\frac{4 \sqrt{5} (3 + 2 \sqrt{5})}{- 11}$

Step 7

Group $3 + 2 \sqrt{5}$ and $\sqrt{5}$ together.

$\frac{4 ((3 + 2 \sqrt{5}) \sqrt{5})}{- 11}$

Step 8

Apply the distributive property.

$\frac{4 (3 \sqrt{5} + 2 \sqrt{5} \sqrt{5})}{- 11}$

Step 9

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

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Step 9.1

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

$\frac{4 (3 \sqrt{5} + 2 ({\sqrt{5}}^{1} \sqrt{5}))}{- 11}$

Step 9.2

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

$\frac{4 (3 \sqrt{5} + 2 ({\sqrt{5}}^{1} {\sqrt{5}}^{1}))}{- 11}$

Step 9.3

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

$\frac{4 (3 \sqrt{5} + 2 {\sqrt{5}}^{1 + 1})}{- 11}$

Step 9.4

Add $1$ and $1$ .

$\frac{4 (3 \sqrt{5} + 2 {\sqrt{5}}^{2})}{- 11}$

Step 10

Simplify each term.

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Step 10.1

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

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Step 10.1.1

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

$\frac{4 (3 \sqrt{5} + 2 {(5^{\frac{1}{2}})}^{2})}{- 11}$

Step 10.1.2

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

$\frac{4 (3 \sqrt{5} + 2 \cdot 5^{\frac{1}{2} \cdot 2})}{- 11}$

Step 10.1.3

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

$\frac{4 (3 \sqrt{5} + 2 \cdot 5^{\frac{2}{2}})}{- 11}$

Step 10.1.4

Cancel the common factor of $2$ .

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Step 10.1.4.1

Cancel the common factor.

$\frac{4 (3 \sqrt{5} + 2 \cdot 5^{\frac{2}{2}})}{- 11}$

Step 10.1.4.2

Rewrite the expression.

$\frac{4 (3 \sqrt{5} + 2 \cdot 5^{1})}{- 11}$

Step 10.1.5

Evaluate the exponent.

$\frac{4 (3 \sqrt{5} + 2 \cdot 5)}{- 11}$

Step 10.2

Multiply $2$ by $5$ .

$\frac{4 (3 \sqrt{5} + 10)}{- 11}$

Step 11

Move the negative in front of the fraction.

$- \frac{4 (3 \sqrt{5} + 10)}{11}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$- \frac{4 (3 \sqrt{5} + 10)}{11}$

Decimal Form:

$- 6.07571052 \dots$