Algebra Examples

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

Step 1

Multiply $\frac{1 + \sqrt{5}}{5 + \sqrt{5}}$ by $\frac{5 - \sqrt{5}}{5 - \sqrt{5}}$ .

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

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{1 + \sqrt{5}}{5 + \sqrt{5}}$ by $\frac{5 - \sqrt{5}}{5 - \sqrt{5}}$ .

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

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(1 + \sqrt{5}) (5 - \sqrt{5})}{25 - 5 \sqrt{5} + \sqrt{5} \cdot 5 - {\sqrt{5}}^{2}}$

Step 2.3

Simplify.

$\frac{(1 + \sqrt{5}) (5 - \sqrt{5})}{20}$

Step 3

Simplify the numerator.

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Step 3.1

Expand $(1 + \sqrt{5}) (5 - \sqrt{5})$ using the FOIL Method.

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Step 3.1.1

Apply the distributive property.

$\frac{1 (5 - \sqrt{5}) + \sqrt{5} (5 - \sqrt{5})}{20}$

Step 3.1.2

Apply the distributive property.

$\frac{1 \cdot 5 + 1 (- \sqrt{5}) + \sqrt{5} (5 - \sqrt{5})}{20}$

Step 3.1.3

Apply the distributive property.

$\frac{1 \cdot 5 + 1 (- \sqrt{5}) + \sqrt{5} \cdot 5 + \sqrt{5} (- \sqrt{5})}{20}$

Step 3.2

Simplify and combine like terms.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Multiply $5$ by $1$ .

$\frac{5 + 1 (- \sqrt{5}) + \sqrt{5} \cdot 5 + \sqrt{5} (- \sqrt{5})}{20}$

Step 3.2.1.2

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

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

Step 3.2.1.3

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

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

Step 3.2.1.4

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

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Step 3.2.1.4.1

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

$\frac{5 - \sqrt{5} + 5 \sqrt{5} - ({\sqrt{5}}^{1} \sqrt{5})}{20}$

Step 3.2.1.4.2

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

$\frac{5 - \sqrt{5} + 5 \sqrt{5} - ({\sqrt{5}}^{1} {\sqrt{5}}^{1})}{20}$

Step 3.2.1.4.3

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

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

Step 3.2.1.4.4

Add $1$ and $1$ .

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

Step 3.2.1.5

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

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Step 3.2.1.5.1

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

$\frac{5 - \sqrt{5} + 5 \sqrt{5} - {(5^{\frac{1}{2}})}^{2}}{20}$

Step 3.2.1.5.2

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

$\frac{5 - \sqrt{5} + 5 \sqrt{5} - 5^{\frac{1}{2} \cdot 2}}{20}$

Step 3.2.1.5.3

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

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

Step 3.2.1.5.4

Cancel the common factor of $2$ .

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Step 3.2.1.5.4.1

Cancel the common factor.

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

Step 3.2.1.5.4.2

Rewrite the expression.

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

Step 3.2.1.5.5

Evaluate the exponent.

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

Step 3.2.1.6

Multiply $- 1$ by $5$ .

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

Step 3.2.2

Subtract $5$ from $5$ .

$\frac{0 - \sqrt{5} + 5 \sqrt{5}}{20}$

Step 3.2.3

Subtract $\sqrt{5}$ from $0$ .

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

Step 3.2.4

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

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

Step 4

Cancel the common factor of $4$ and $20$ .

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Step 4.1

Factor $4$ out of $4 \sqrt{5}$ .

$\frac{4 (\sqrt{5})}{20}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $4$ out of $20$ .

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.44721359 \dots$