Basic Math Examples

\frac{\sqrt{22} - \sqrt{5}}{\sqrt{22} + \sqrt{5}}

$\frac{\sqrt{22} - \sqrt{5}}{\sqrt{22} + \sqrt{5}}$

Step 1

Multiply

$\frac{\sqrt{22} - \sqrt{5}}{\sqrt{22} + \sqrt{5}}$ by

$\frac{\sqrt{22} - \sqrt{5}}{\sqrt{22} - \sqrt{5}}$ .

$\frac{\sqrt{22} - \sqrt{5}}{\sqrt{22} + \sqrt{5}} \cdot \frac{\sqrt{22} - \sqrt{5}}{\sqrt{22} - \sqrt{5}}$

Step 2

Combine fractions.

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Step 2.1

Multiply

$\frac{\sqrt{22} - \sqrt{5}}{\sqrt{22} + \sqrt{5}}$ by

$\frac{\sqrt{22} - \sqrt{5}}{\sqrt{22} - \sqrt{5}}$ .

$\frac{(\sqrt{22} - \sqrt{5}) (\sqrt{22} - \sqrt{5})}{(\sqrt{22} + \sqrt{5}) (\sqrt{22} - \sqrt{5})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(\sqrt{22} - \sqrt{5}) (\sqrt{22} - \sqrt{5})}{{\sqrt{22}}^{2} - \sqrt{110} + \sqrt{110} - {\sqrt{5}}^{2}}$

Step 2.3

Simplify.

$\frac{(\sqrt{22} - \sqrt{5}) (\sqrt{22} - \sqrt{5})}{17}$

Step 3

Simplify the numerator.

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Step 3.1

Raise

$\sqrt{22} - \sqrt{5}$ to the power of

$1$ .

$\frac{{(\sqrt{22} - \sqrt{5})}^{1} (\sqrt{22} - \sqrt{5})}{17}$

Step 3.2

Raise

$\sqrt{22} - \sqrt{5}$ to the power of

$1$ .

$\frac{{(\sqrt{22} - \sqrt{5})}^{1} {(\sqrt{22} - \sqrt{5})}^{1}}{17}$

Step 3.3

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{(\sqrt{22} - \sqrt{5})}^{1 + 1}}{17}$

Step 3.4

Add

$1$ and

$1$ .

$\frac{{(\sqrt{22} - \sqrt{5})}^{2}}{17}$

Step 4

Rewrite

${(\sqrt{22} - \sqrt{5})}^{2}$ as

$(\sqrt{22} - \sqrt{5}) (\sqrt{22} - \sqrt{5})$ .

$\frac{(\sqrt{22} - \sqrt{5}) (\sqrt{22} - \sqrt{5})}{17}$

Step 5

Expand

$(\sqrt{22} - \sqrt{5}) (\sqrt{22} - \sqrt{5})$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$\frac{\sqrt{22} (\sqrt{22} - \sqrt{5}) - \sqrt{5} (\sqrt{22} - \sqrt{5})}{17}$

Step 5.2

Apply the distributive property.

$\frac{\sqrt{22} \sqrt{22} + \sqrt{22} (- \sqrt{5}) - \sqrt{5} (\sqrt{22} - \sqrt{5})}{17}$

Step 5.3

Apply the distributive property.

$\frac{\sqrt{22} \sqrt{22} + \sqrt{22} (- \sqrt{5}) - \sqrt{5} \sqrt{22} - \sqrt{5} (- \sqrt{5})}{17}$

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

Combine using the product rule for radicals.

$\frac{\sqrt{22 \cdot 22} + \sqrt{22} (- \sqrt{5}) - \sqrt{5} \sqrt{22} - \sqrt{5} (- \sqrt{5})}{17}$

Step 6.1.2

Multiply

$22$ by

$22$ .

$\frac{\sqrt{484} + \sqrt{22} (- \sqrt{5}) - \sqrt{5} \sqrt{22} - \sqrt{5} (- \sqrt{5})}{17}$

Step 6.1.3

Rewrite

$484$ as

$22^{2}$ .

$\frac{\sqrt{22^{2}} + \sqrt{22} (- \sqrt{5}) - \sqrt{5} \sqrt{22} - \sqrt{5} (- \sqrt{5})}{17}$

Step 6.1.4

Pull terms out from under the radical, assuming positive real numbers.

$\frac{22 + \sqrt{22} (- \sqrt{5}) - \sqrt{5} \sqrt{22} - \sqrt{5} (- \sqrt{5})}{17}$

Step 6.1.5

Multiply

$\sqrt{22} (- \sqrt{5})$ .

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Step 6.1.5.1

Combine using the product rule for radicals.

$\frac{22 - \sqrt{22 \cdot 5} - \sqrt{5} \sqrt{22} - \sqrt{5} (- \sqrt{5})}{17}$

Step 6.1.5.2

Multiply

$22$ by

$5$ .

$\frac{22 - \sqrt{110} - \sqrt{5} \sqrt{22} - \sqrt{5} (- \sqrt{5})}{17}$

Step 6.1.6

Multiply

$- \sqrt{5} \sqrt{22}$ .

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Step 6.1.6.1

Combine using the product rule for radicals.

$\frac{22 - \sqrt{110} - \sqrt{22 \cdot 5} - \sqrt{5} (- \sqrt{5})}{17}$

Step 6.1.6.2

Multiply

$22$ by

$5$ .

$\frac{22 - \sqrt{110} - \sqrt{110} - \sqrt{5} (- \sqrt{5})}{17}$

Step 6.1.7

Multiply

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

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Step 6.1.7.1

Multiply

$- 1$ by

$- 1$ .

$\frac{22 - \sqrt{110} - \sqrt{110} + 1 \sqrt{5} \sqrt{5}}{17}$

Step 6.1.7.2

Multiply

$\sqrt{5}$ by

$1$ .

$\frac{22 - \sqrt{110} - \sqrt{110} + \sqrt{5} \sqrt{5}}{17}$

Step 6.1.7.3

Raise

$\sqrt{5}$ to the power of

$1$ .

$\frac{22 - \sqrt{110} - \sqrt{110} + {\sqrt{5}}^{1} \sqrt{5}}{17}$

Step 6.1.7.4

Raise

$\sqrt{5}$ to the power of

$1$ .

$\frac{22 - \sqrt{110} - \sqrt{110} + {\sqrt{5}}^{1} {\sqrt{5}}^{1}}{17}$

Step 6.1.7.5

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{22 - \sqrt{110} - \sqrt{110} + {\sqrt{5}}^{1 + 1}}{17}$

Step 6.1.7.6

Add

$1$ and

$1$ .

$\frac{22 - \sqrt{110} - \sqrt{110} + {\sqrt{5}}^{2}}{17}$

Step 6.1.8

Rewrite

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

$5$ .

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Step 6.1.8.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{5}$ as

$5^{\frac{1}{2}}$ .

$\frac{22 - \sqrt{110} - \sqrt{110} + {(5^{\frac{1}{2}})}^{2}}{17}$

Step 6.1.8.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\frac{22 - \sqrt{110} - \sqrt{110} + 5^{\frac{1}{2} \cdot 2}}{17}$

Step 6.1.8.3

Combine

$\frac{1}{2}$ and

$2$ .

$\frac{22 - \sqrt{110} - \sqrt{110} + 5^{\frac{2}{2}}}{17}$

Step 6.1.8.4

Cancel the common factor of

$2$ .

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Step 6.1.8.4.1

Cancel the common factor.

$\frac{22 - \sqrt{110} - \sqrt{110} + 5^{\frac{2}{2}}}{17}$

Step 6.1.8.4.2

Rewrite the expression.

$\frac{22 - \sqrt{110} - \sqrt{110} + 5^{1}}{17}$

Step 6.1.8.5

Evaluate the exponent.

$\frac{22 - \sqrt{110} - \sqrt{110} + 5}{17}$

Step 6.2

Add

$22$ and

$5$ .

$\frac{27 - \sqrt{110} - \sqrt{110}}{17}$

Step 6.3

Subtract

$\sqrt{110}$ from

$- \sqrt{110}$ .

$\frac{27 - 2 \sqrt{110}}{17}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{27 - 2 \sqrt{110}}{17}$

Decimal Form:

$0.35434253 \dots$