Simplify (7- square root of 5)/(4+ square root of 5)

$\frac{7 - \sqrt{5}}{4 + \sqrt{5}}$

Multiply $\frac{7 - \sqrt{5}}{4 + \sqrt{5}}$ by $\frac{4 - \sqrt{5}}{4 - \sqrt{5}}$ .

$\frac{7 - \sqrt{5}}{4 + \sqrt{5}} \cdot \frac{4 - \sqrt{5}}{4 - \sqrt{5}}$

Combine fractions.

Multiply $\frac{7 - \sqrt{5}}{4 + \sqrt{5}}$ by $\frac{4 - \sqrt{5}}{4 - \sqrt{5}}$ .

$\frac{(7 - \sqrt{5}) (4 - \sqrt{5})}{(4 + \sqrt{5}) (4 - \sqrt{5})}$

Expand the denominator using the FOIL method.

$\frac{(7 - \sqrt{5}) (4 - \sqrt{5})}{16 - 4 \sqrt{5} + \sqrt{5} \cdot 4 - {\sqrt{5}}^{2}}$

Simplify.

$\frac{(7 - \sqrt{5}) (4 - \sqrt{5})}{11}$

Expand $(7 - \sqrt{5}) (4 - \sqrt{5})$ using the FOIL Method.

Apply the distributive property.

$\frac{7 (4 - \sqrt{5}) - \sqrt{5} (4 - \sqrt{5})}{11}$

Apply the distributive property.

$\frac{7 \cdot 4 + 7 (- \sqrt{5}) - \sqrt{5} (4 - \sqrt{5})}{11}$

Apply the distributive property.

$\frac{7 \cdot 4 + 7 (- \sqrt{5}) - \sqrt{5} \cdot 4 - \sqrt{5} (- \sqrt{5})}{11}$

Simplify and combine like terms.

Simplify each term.

Multiply $7$ by $4$ .

$\frac{28 + 7 (- \sqrt{5}) - \sqrt{5} \cdot 4 - \sqrt{5} (- \sqrt{5})}{11}$

Multiply $- 1$ by $7$ .

$\frac{28 - 7 \sqrt{5} - \sqrt{5} \cdot 4 - \sqrt{5} (- \sqrt{5})}{11}$

Multiply $4$ by $- 1$ .

$\frac{28 - 7 \sqrt{5} - 4 \sqrt{5} - \sqrt{5} (- \sqrt{5})}{11}$

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

Multiply $- 1$ by $- 1$ .

$\frac{28 - 7 \sqrt{5} - 4 \sqrt{5} + 1 \sqrt{5} \sqrt{5}}{11}$

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

$\frac{28 - 7 \sqrt{5} - 4 \sqrt{5} + \sqrt{5} \sqrt{5}}{11}$

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

$\frac{28 - 7 \sqrt{5} - 4 \sqrt{5} + {\sqrt{5}}^{1} \sqrt{5}}{11}$

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

$\frac{28 - 7 \sqrt{5} - 4 \sqrt{5} + {\sqrt{5}}^{1} {\sqrt{5}}^{1}}{11}$

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

$\frac{28 - 7 \sqrt{5} - 4 \sqrt{5} + {\sqrt{5}}^{1 + 1}}{11}$

Add $1$ and $1$ .

$\frac{28 - 7 \sqrt{5} - 4 \sqrt{5} + {\sqrt{5}}^{2}}{11}$

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

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

$\frac{28 - 7 \sqrt{5} - 4 \sqrt{5} + {(5^{\frac{1}{2}})}^{2}}{11}$

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

$\frac{28 - 7 \sqrt{5} - 4 \sqrt{5} + 5^{\frac{1}{2} \cdot 2}}{11}$

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

$\frac{28 - 7 \sqrt{5} - 4 \sqrt{5} + 5^{\frac{2}{2}}}{11}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{28 - 7 \sqrt{5} - 4 \sqrt{5} + 5^{\frac{2}{2}}}{11}$

Rewrite the expression.

$\frac{28 - 7 \sqrt{5} - 4 \sqrt{5} + 5^{1}}{11}$

Evaluate the exponent.

$\frac{28 - 7 \sqrt{5} - 4 \sqrt{5} + 5}{11}$

Add $28$ and $5$ .

$\frac{33 - 7 \sqrt{5} - 4 \sqrt{5}}{11}$

Subtract $4 \sqrt{5}$ from $- 7 \sqrt{5}$ .

$\frac{33 - 11 \sqrt{5}}{11}$

Cancel the common factor of $33 - 11 \sqrt{5}$ and $11$ .

Factor $11$ out of $33$ .

$\frac{11 \cdot 3 - 11 \sqrt{5}}{11}$

Factor $11$ out of $- 11 \sqrt{5}$ .

$\frac{11 \cdot 3 + 11 (- \sqrt{5})}{11}$

Factor $11$ out of $11 (3) + 11 (- \sqrt{5})$ .

$\frac{11 (3 - \sqrt{5})}{11}$

Cancel the common factors.

Factor $11$ out of $11$ .

$\frac{11 (3 - \sqrt{5})}{11 (1)}$

Cancel the common factor.

$\frac{11 (3 - \sqrt{5})}{11 \cdot 1}$

Rewrite the expression.

$\frac{3 - \sqrt{5}}{1}$

Divide $3 - \sqrt{5}$ by $1$ .

$3 - \sqrt{5}$

The result can be shown in multiple forms.

Exact Form:

$3 - \sqrt{5}$

Decimal Form:

$0.76393202 \dots$

Simplify (7- square root of 5)/(4+ square root of 5)

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