Simplify (1+ square root of 5)/(5+ square root of 5)

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

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

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

Combine fractions.

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

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

Expand the denominator using the FOIL method.

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

Simplify.

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

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $1$ .

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

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

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

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Multiply $- 1$ by $5$ .

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

Subtract $5$ from $5$ .

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

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

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

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

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

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

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

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

Cancel the common factors.

Factor $4$ out of $20$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.44721359 \dots$

Simplify (1+ square root of 5)/(5+ square root of 5)

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