Algebra Examples

$\frac{37 + 12 \sqrt{5}}{\sqrt{5} + 8}$

Step 1

Multiply $\frac{37 + 12 \sqrt{5}}{\sqrt{5} + 8}$ by $\frac{\sqrt{5} - 8}{\sqrt{5} - 8}$ .

$\frac{37 + 12 \sqrt{5}}{\sqrt{5} + 8} \cdot \frac{\sqrt{5} - 8}{\sqrt{5} - 8}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{37 + 12 \sqrt{5}}{\sqrt{5} + 8}$ by $\frac{\sqrt{5} - 8}{\sqrt{5} - 8}$ .

$\frac{(37 + 12 \sqrt{5}) (\sqrt{5} - 8)}{(\sqrt{5} + 8) (\sqrt{5} - 8)}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(37 + 12 \sqrt{5}) (\sqrt{5} - 8)}{{\sqrt{5}}^{2} + \sqrt{5} \cdot - 8 + 8 \sqrt{5} - 64}$

Step 2.3

Simplify.

$\frac{(37 + 12 \sqrt{5}) (\sqrt{5} - 8)}{- 59}$

Step 3

Expand $(37 + 12 \sqrt{5}) (\sqrt{5} - 8)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{37 (\sqrt{5} - 8) + 12 \sqrt{5} (\sqrt{5} - 8)}{- 59}$

Step 3.2

Apply the distributive property.

$\frac{37 \sqrt{5} + 37 \cdot - 8 + 12 \sqrt{5} (\sqrt{5} - 8)}{- 59}$

Step 3.3

Apply the distributive property.

$\frac{37 \sqrt{5} + 37 \cdot - 8 + 12 \sqrt{5} \sqrt{5} + 12 \sqrt{5} \cdot - 8}{- 59}$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $37$ by $- 8$ .

$\frac{37 \sqrt{5} - 296 + 12 \sqrt{5} \sqrt{5} + 12 \sqrt{5} \cdot - 8}{- 59}$

Step 4.1.2

Multiply $12 \sqrt{5} \sqrt{5}$ .

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Step 4.1.2.1

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

$\frac{37 \sqrt{5} - 296 + 12 ({\sqrt{5}}^{1} \sqrt{5}) + 12 \sqrt{5} \cdot - 8}{- 59}$

Step 4.1.2.2

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

$\frac{37 \sqrt{5} - 296 + 12 ({\sqrt{5}}^{1} {\sqrt{5}}^{1}) + 12 \sqrt{5} \cdot - 8}{- 59}$

Step 4.1.2.3

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

$\frac{37 \sqrt{5} - 296 + 12 {\sqrt{5}}^{1 + 1} + 12 \sqrt{5} \cdot - 8}{- 59}$

Step 4.1.2.4

Add $1$ and $1$ .

$\frac{37 \sqrt{5} - 296 + 12 {\sqrt{5}}^{2} + 12 \sqrt{5} \cdot - 8}{- 59}$

Step 4.1.3

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

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Step 4.1.3.1

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

$\frac{37 \sqrt{5} - 296 + 12 {(5^{\frac{1}{2}})}^{2} + 12 \sqrt{5} \cdot - 8}{- 59}$

Step 4.1.3.2

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

$\frac{37 \sqrt{5} - 296 + 12 \cdot 5^{\frac{1}{2} \cdot 2} + 12 \sqrt{5} \cdot - 8}{- 59}$

Step 4.1.3.3

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

$\frac{37 \sqrt{5} - 296 + 12 \cdot 5^{\frac{2}{2}} + 12 \sqrt{5} \cdot - 8}{- 59}$

Step 4.1.3.4

Cancel the common factor of $2$ .

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Step 4.1.3.4.1

Cancel the common factor.

$\frac{37 \sqrt{5} - 296 + 12 \cdot 5^{\frac{2}{2}} + 12 \sqrt{5} \cdot - 8}{- 59}$

Step 4.1.3.4.2

Rewrite the expression.

$\frac{37 \sqrt{5} - 296 + 12 \cdot 5^{1} + 12 \sqrt{5} \cdot - 8}{- 59}$

Step 4.1.3.5

Evaluate the exponent.

$\frac{37 \sqrt{5} - 296 + 12 \cdot 5 + 12 \sqrt{5} \cdot - 8}{- 59}$

Step 4.1.4

Multiply $12$ by $5$ .

$\frac{37 \sqrt{5} - 296 + 60 + 12 \sqrt{5} \cdot - 8}{- 59}$

Step 4.1.5

Multiply $- 8$ by $12$ .

$\frac{37 \sqrt{5} - 296 + 60 - 96 \sqrt{5}}{- 59}$

Step 4.2

Subtract $96 \sqrt{5}$ from $37 \sqrt{5}$ .

$\frac{- 296 + 60 - 59 \sqrt{5}}{- 59}$

Step 4.3

Add $- 296$ and $60$ .

$\frac{- 236 - 59 \sqrt{5}}{- 59}$

Step 5

Cancel the common factor of $- 236 - 59 \sqrt{5}$ and $- 59$ .

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Step 5.1

Factor $59$ out of $- 236$ .

$\frac{59 (- 4) - 59 \sqrt{5}}{- 59}$

Step 5.2

Factor $59$ out of $- 59 \sqrt{5}$ .

$\frac{59 (- 4) + 59 (- \sqrt{5})}{- 59}$

Step 5.3

Factor $59$ out of $59 (- 4) + 59 (- \sqrt{5})$ .

$\frac{59 (- 4 - \sqrt{5})}{- 59}$

Step 5.4

Move the negative one from the denominator of $\frac{- 4 - \sqrt{5}}{- 1}$ .

$- 1 \cdot (- 4 - \sqrt{5})$

Step 6

Rewrite $- 1 \cdot (- 4 - \sqrt{5})$ as $- (- 4 - \sqrt{5})$ .

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

Step 7

Apply the distributive property.

$- - 4 - - \sqrt{5}$

Step 8

Multiply $- 1$ by $- 4$ .

$4 - - \sqrt{5}$

Step 9

Multiply $- - \sqrt{5}$ .

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Step 9.1

Multiply $- 1$ by $- 1$ .

$4 + 1 \sqrt{5}$

Step 9.2

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

$4 + \sqrt{5}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$4 + \sqrt{5}$

Decimal Form:

$6.23606797 \dots$