Algebra Examples

$\frac{6 - \sqrt{5}}{6 + \sqrt{5}}$

Step 1

Multiply $\frac{6 - \sqrt{5}}{6 + \sqrt{5}}$ by $\frac{6 - \sqrt{5}}{6 - \sqrt{5}}$ .

$\frac{6 - \sqrt{5}}{6 + \sqrt{5}} \cdot \frac{6 - \sqrt{5}}{6 - \sqrt{5}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{6 - \sqrt{5}}{6 + \sqrt{5}}$ by $\frac{6 - \sqrt{5}}{6 - \sqrt{5}}$ .

$\frac{(6 - \sqrt{5}) (6 - \sqrt{5})}{(6 + \sqrt{5}) (6 - \sqrt{5})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(6 - \sqrt{5}) (6 - \sqrt{5})}{36 - 6 \sqrt{5} + \sqrt{5} \cdot 6 - {\sqrt{5}}^{2}}$

Step 2.3

Simplify.

$\frac{(6 - \sqrt{5}) (6 - \sqrt{5})}{31}$

Step 3

Simplify the numerator.

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Step 3.1

Raise $6 - \sqrt{5}$ to the power of $1$ .

$\frac{{(6 - \sqrt{5})}^{1} (6 - \sqrt{5})}{31}$

Step 3.2

Raise $6 - \sqrt{5}$ to the power of $1$ .

$\frac{{(6 - \sqrt{5})}^{1} {(6 - \sqrt{5})}^{1}}{31}$

Step 3.3

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

$\frac{{(6 - \sqrt{5})}^{1 + 1}}{31}$

Step 3.4

Add $1$ and $1$ .

$\frac{{(6 - \sqrt{5})}^{2}}{31}$

Step 4

Rewrite ${(6 - \sqrt{5})}^{2}$ as $(6 - \sqrt{5}) (6 - \sqrt{5})$ .

$\frac{(6 - \sqrt{5}) (6 - \sqrt{5})}{31}$

Step 5

Expand $(6 - \sqrt{5}) (6 - \sqrt{5})$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$\frac{6 (6 - \sqrt{5}) - \sqrt{5} (6 - \sqrt{5})}{31}$

Step 5.2

Apply the distributive property.

$\frac{6 \cdot 6 + 6 (- \sqrt{5}) - \sqrt{5} (6 - \sqrt{5})}{31}$

Step 5.3

Apply the distributive property.

$\frac{6 \cdot 6 + 6 (- \sqrt{5}) - \sqrt{5} \cdot 6 - \sqrt{5} (- \sqrt{5})}{31}$

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $6$ by $6$ .

$\frac{36 + 6 (- \sqrt{5}) - \sqrt{5} \cdot 6 - \sqrt{5} (- \sqrt{5})}{31}$

Step 6.1.2

Multiply $- 1$ by $6$ .

$\frac{36 - 6 \sqrt{5} - \sqrt{5} \cdot 6 - \sqrt{5} (- \sqrt{5})}{31}$

Step 6.1.3

Multiply $6$ by $- 1$ .

$\frac{36 - 6 \sqrt{5} - 6 \sqrt{5} - \sqrt{5} (- \sqrt{5})}{31}$

Step 6.1.4

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

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Step 6.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{36 - 6 \sqrt{5} - 6 \sqrt{5} + 1 \sqrt{5} \sqrt{5}}{31}$

Step 6.1.4.2

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

$\frac{36 - 6 \sqrt{5} - 6 \sqrt{5} + \sqrt{5} \sqrt{5}}{31}$

Step 6.1.4.3

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

$\frac{36 - 6 \sqrt{5} - 6 \sqrt{5} + {\sqrt{5}}^{1} \sqrt{5}}{31}$

Step 6.1.4.4

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

$\frac{36 - 6 \sqrt{5} - 6 \sqrt{5} + {\sqrt{5}}^{1} {\sqrt{5}}^{1}}{31}$

Step 6.1.4.5

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

$\frac{36 - 6 \sqrt{5} - 6 \sqrt{5} + {\sqrt{5}}^{1 + 1}}{31}$

Step 6.1.4.6

Add $1$ and $1$ .

$\frac{36 - 6 \sqrt{5} - 6 \sqrt{5} + {\sqrt{5}}^{2}}{31}$

Step 6.1.5

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

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Step 6.1.5.1

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

$\frac{36 - 6 \sqrt{5} - 6 \sqrt{5} + {(5^{\frac{1}{2}})}^{2}}{31}$

Step 6.1.5.2

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

$\frac{36 - 6 \sqrt{5} - 6 \sqrt{5} + 5^{\frac{1}{2} \cdot 2}}{31}$

Step 6.1.5.3

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

$\frac{36 - 6 \sqrt{5} - 6 \sqrt{5} + 5^{\frac{2}{2}}}{31}$

Step 6.1.5.4

Cancel the common factor of $2$ .

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Step 6.1.5.4.1

Cancel the common factor.

$\frac{36 - 6 \sqrt{5} - 6 \sqrt{5} + 5^{\frac{2}{2}}}{31}$

Step 6.1.5.4.2

Rewrite the expression.

$\frac{36 - 6 \sqrt{5} - 6 \sqrt{5} + 5^{1}}{31}$

Step 6.1.5.5

Evaluate the exponent.

$\frac{36 - 6 \sqrt{5} - 6 \sqrt{5} + 5}{31}$

Step 6.2

Add $36$ and $5$ .

$\frac{41 - 6 \sqrt{5} - 6 \sqrt{5}}{31}$

Step 6.3

Subtract $6 \sqrt{5}$ from $- 6 \sqrt{5}$ .

$\frac{41 - 12 \sqrt{5}}{31}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{41 - 12 \sqrt{5}}{31}$

Decimal Form:

$0.45700594 \dots$