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

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

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

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

Combine fractions.

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

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

Expand the denominator using the FOIL method.

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

Simplify.

$\frac{(5 - \sqrt{5}) (1 + \sqrt{5})}{- 4}$

Simplify the numerator.

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

Apply the distributive property.

$\frac{5 (1 + \sqrt{5}) - \sqrt{5} (1 + \sqrt{5})}{- 4}$

Apply the distributive property.

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

Apply the distributive property.

$\frac{5 \cdot 1 + 5 \sqrt{5} - \sqrt{5} \cdot 1 - \sqrt{5} \sqrt{5}}{- 4}$

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $1$ .

$\frac{5 + 5 \sqrt{5} - \sqrt{5} \cdot 1 - \sqrt{5} \sqrt{5}}{- 4}$

Multiply $- 1$ by $1$ .

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

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

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

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

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

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

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

$\frac{5 + 5 \sqrt{5} - \sqrt{5} - {\sqrt{5}}^{1 + 1}}{- 4}$

Add $1$ and $1$ .

$\frac{5 + 5 \sqrt{5} - \sqrt{5} - {\sqrt{5}}^{2}}{- 4}$

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

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

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

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

$\frac{5 + 5 \sqrt{5} - \sqrt{5} - 5^{\frac{1}{2} \cdot 2}}{- 4}$

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

$\frac{5 + 5 \sqrt{5} - \sqrt{5} - 5^{\frac{2}{2}}}{- 4}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{5 + 5 \sqrt{5} - \sqrt{5} - 5^{\frac{2}{2}}}{- 4}$

Rewrite the expression.

$\frac{5 + 5 \sqrt{5} - \sqrt{5} - 5^{1}}{- 4}$

Evaluate the exponent.

$\frac{5 + 5 \sqrt{5} - \sqrt{5} - 1 \cdot 5}{- 4}$

Multiply $- 1$ by $5$ .

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

Subtract $5$ from $5$ .

$\frac{0 + 5 \sqrt{5} - \sqrt{5}}{- 4}$

Add $0$ and $5 \sqrt{5}$ .

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

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

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

Reduce the expression by cancelling the common factors.

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

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

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

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

$- 1 \cdot \sqrt{5}$

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

$- \sqrt{5}$

The result can be shown in multiple forms.

Exact Form:

$- \sqrt{5}$

Decimal Form:

$- 2.23606797 \dots$

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

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