Algebra Examples

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

Step 1

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

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

Step 2

Combine fractions.

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Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

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

Step 3

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

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $7$ by $4$ .

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

Step 4.1.2

Multiply $- 1$ by $7$ .

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

Step 4.1.3

Multiply $4$ by $- 1$ .

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

Step 4.1.4

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

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Step 4.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 4.1.4.2

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

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

Step 4.1.4.3

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

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

Step 4.1.4.4

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

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

Step 4.1.4.5

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

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

Step 4.1.4.6

Add $1$ and $1$ .

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

Step 4.1.5

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

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Step 4.1.5.1

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

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

Step 4.1.5.2

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

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

Step 4.1.5.3

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

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

Step 4.1.5.4

Cancel the common factor of $2$ .

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Step 4.1.5.4.1

Cancel the common factor.

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

Step 4.1.5.4.2

Rewrite the expression.

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

Step 4.1.5.5

Evaluate the exponent.

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

Step 4.2

Add $28$ and $5$ .

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

Step 4.3

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

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

Step 5

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

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Step 5.1

Factor $11$ out of $33$ .

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Cancel the common factors.

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Step 5.4.1

Factor $11$ out of $11$ .

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

Step 5.4.2

Cancel the common factor.

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

Step 5.4.3

Rewrite the expression.

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

Step 5.4.4

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

$3 - \sqrt{5}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$3 - \sqrt{5}$

Decimal Form:

$0.76393202 \dots$