Pre-Algebra Examples

$\frac{2 + \sqrt{5}}{6 + 2 \sqrt{5}}$

Step 1

Multiply $\frac{2 + \sqrt{5}}{6 + 2 \sqrt{5}}$ by $\frac{6 - 2 \sqrt{5}}{6 - 2 \sqrt{5}}$ .

$\frac{2 + \sqrt{5}}{6 + 2 \sqrt{5}} \cdot \frac{6 - 2 \sqrt{5}}{6 - 2 \sqrt{5}}$

Step 2

Multiply $\frac{2 + \sqrt{5}}{6 + 2 \sqrt{5}}$ by $\frac{6 - 2 \sqrt{5}}{6 - 2 \sqrt{5}}$ .

$\frac{(2 + \sqrt{5}) (6 - 2 \sqrt{5})}{(6 + 2 \sqrt{5}) (6 - 2 \sqrt{5})}$

Step 3

Expand the denominator using the FOIL method.

$\frac{(2 + \sqrt{5}) (6 - 2 \sqrt{5})}{36 - 12 \sqrt{5} + 12 \sqrt{5} - 4 {\sqrt{5}}^{2}}$

Step 4

Simplify.

$\frac{(2 + \sqrt{5}) (6 - 2 \sqrt{5})}{16}$

Step 5

Cancel the common factor of $6 - 2 \sqrt{5}$ and $16$ .

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Step 5.1

Factor $2$ out of $(2 + \sqrt{5}) (6 - 2 \sqrt{5})$ .

$\frac{2 ((2 + \sqrt{5}) (3 - \sqrt{5}))}{16}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $2$ out of $16$ .

$\frac{2 ((2 + \sqrt{5}) (3 - \sqrt{5}))}{2 (8)}$

Step 5.2.2

Cancel the common factor.

$\frac{2 ((2 + \sqrt{5}) (3 - \sqrt{5}))}{2 \cdot 8}$

Step 5.2.3

Rewrite the expression.

$\frac{(2 + \sqrt{5}) (3 - \sqrt{5})}{8}$

Step 6

Expand $(2 + \sqrt{5}) (3 - \sqrt{5})$ using the FOIL Method.

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Step 6.1

Apply the distributive property.

$\frac{2 (3 - \sqrt{5}) + \sqrt{5} (3 - \sqrt{5})}{8}$

Step 6.2

Apply the distributive property.

$\frac{2 \cdot 3 + 2 (- \sqrt{5}) + \sqrt{5} (3 - \sqrt{5})}{8}$

Step 6.3

Apply the distributive property.

$\frac{2 \cdot 3 + 2 (- \sqrt{5}) + \sqrt{5} \cdot 3 + \sqrt{5} (- \sqrt{5})}{8}$

Step 7

Simplify and combine like terms.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply $2$ by $3$ .

$\frac{6 + 2 (- \sqrt{5}) + \sqrt{5} \cdot 3 + \sqrt{5} (- \sqrt{5})}{8}$

Step 7.1.2

Multiply $- 1$ by $2$ .

$\frac{6 - 2 \sqrt{5} + \sqrt{5} \cdot 3 + \sqrt{5} (- \sqrt{5})}{8}$

Step 7.1.3

Move $3$ to the left of $\sqrt{5}$ .

$\frac{6 - 2 \sqrt{5} + 3 \cdot \sqrt{5} + \sqrt{5} (- \sqrt{5})}{8}$

Step 7.1.4

Multiply $\sqrt{5} (- \sqrt{5})$ .

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Step 7.1.4.1

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

$\frac{6 - 2 \sqrt{5} + 3 \sqrt{5} - ({\sqrt{5}}^{1} \sqrt{5})}{8}$

Step 7.1.4.2

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

$\frac{6 - 2 \sqrt{5} + 3 \sqrt{5} - ({\sqrt{5}}^{1} {\sqrt{5}}^{1})}{8}$

Step 7.1.4.3

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

$\frac{6 - 2 \sqrt{5} + 3 \sqrt{5} - {\sqrt{5}}^{1 + 1}}{8}$

Step 7.1.4.4

Add $1$ and $1$ .

$\frac{6 - 2 \sqrt{5} + 3 \sqrt{5} - {\sqrt{5}}^{2}}{8}$

Step 7.1.5

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

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Step 7.1.5.1

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

$\frac{6 - 2 \sqrt{5} + 3 \sqrt{5} - {(5^{\frac{1}{2}})}^{2}}{8}$

Step 7.1.5.2

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

$\frac{6 - 2 \sqrt{5} + 3 \sqrt{5} - 5^{\frac{1}{2} \cdot 2}}{8}$

Step 7.1.5.3

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

$\frac{6 - 2 \sqrt{5} + 3 \sqrt{5} - 5^{\frac{2}{2}}}{8}$

Step 7.1.5.4

Cancel the common factor of $2$ .

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Step 7.1.5.4.1

Cancel the common factor.

$\frac{6 - 2 \sqrt{5} + 3 \sqrt{5} - 5^{\frac{2}{2}}}{8}$

Step 7.1.5.4.2

Rewrite the expression.

$\frac{6 - 2 \sqrt{5} + 3 \sqrt{5} - 5^{1}}{8}$

Step 7.1.5.5

Evaluate the exponent.

$\frac{6 - 2 \sqrt{5} + 3 \sqrt{5} - 1 \cdot 5}{8}$

Step 7.1.6

Multiply $- 1$ by $5$ .

$\frac{6 - 2 \sqrt{5} + 3 \sqrt{5} - 5}{8}$

Step 7.2

Subtract $5$ from $6$ .

$\frac{1 - 2 \sqrt{5} + 3 \sqrt{5}}{8}$

Step 7.3

Add $- 2 \sqrt{5}$ and $3 \sqrt{5}$ .

$\frac{1 + \sqrt{5}}{8}$

Step 8

Split the fraction $\frac{1 + \sqrt{5}}{8}$ into two fractions.

$\frac{1}{8} + \frac{\sqrt{5}}{8}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{8} + \frac{\sqrt{5}}{8}$

Decimal Form:

$0.40450849 \dots$