Pre-Algebra Examples

$\frac{\sqrt{11} - \sqrt{2}}{\sqrt{11} + \sqrt{2}}$

Step 1

Multiply $\frac{\sqrt{11} - \sqrt{2}}{\sqrt{11} + \sqrt{2}}$ by $\frac{\sqrt{11} - \sqrt{2}}{\sqrt{11} - \sqrt{2}}$ .

$\frac{\sqrt{11} - \sqrt{2}}{\sqrt{11} + \sqrt{2}} \cdot \frac{\sqrt{11} - \sqrt{2}}{\sqrt{11} - \sqrt{2}}$

Step 2

Multiply $\frac{\sqrt{11} - \sqrt{2}}{\sqrt{11} + \sqrt{2}}$ by $\frac{\sqrt{11} - \sqrt{2}}{\sqrt{11} - \sqrt{2}}$ .

$\frac{(\sqrt{11} - \sqrt{2}) (\sqrt{11} - \sqrt{2})}{(\sqrt{11} + \sqrt{2}) (\sqrt{11} - \sqrt{2})}$

Step 3

Expand the denominator using the FOIL method.

$\frac{(\sqrt{11} - \sqrt{2}) (\sqrt{11} - \sqrt{2})}{{\sqrt{11}}^{2} - \sqrt{22} + \sqrt{22} - {\sqrt{2}}^{2}}$

Step 4

Simplify.

$\frac{(\sqrt{11} - \sqrt{2}) (\sqrt{11} - \sqrt{2})}{9}$

Step 5

Simplify the numerator.

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Step 5.1

Raise $\sqrt{11} - \sqrt{2}$ to the power of $1$ .

$\frac{{(\sqrt{11} - \sqrt{2})}^{1} (\sqrt{11} - \sqrt{2})}{9}$

Step 5.2

Raise $\sqrt{11} - \sqrt{2}$ to the power of $1$ .

$\frac{{(\sqrt{11} - \sqrt{2})}^{1} {(\sqrt{11} - \sqrt{2})}^{1}}{9}$

Step 5.3

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

$\frac{{(\sqrt{11} - \sqrt{2})}^{1 + 1}}{9}$

Step 5.4

Add $1$ and $1$ .

$\frac{{(\sqrt{11} - \sqrt{2})}^{2}}{9}$

Step 6

Rewrite ${(\sqrt{11} - \sqrt{2})}^{2}$ as $(\sqrt{11} - \sqrt{2}) (\sqrt{11} - \sqrt{2})$ .

$\frac{(\sqrt{11} - \sqrt{2}) (\sqrt{11} - \sqrt{2})}{9}$

Step 7

Expand $(\sqrt{11} - \sqrt{2}) (\sqrt{11} - \sqrt{2})$ using the FOIL Method.

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Step 7.1

Apply the distributive property.

$\frac{\sqrt{11} (\sqrt{11} - \sqrt{2}) - \sqrt{2} (\sqrt{11} - \sqrt{2})}{9}$

Step 7.2

Apply the distributive property.

$\frac{\sqrt{11} \sqrt{11} + \sqrt{11} (- \sqrt{2}) - \sqrt{2} (\sqrt{11} - \sqrt{2})}{9}$

Step 7.3

Apply the distributive property.

$\frac{\sqrt{11} \sqrt{11} + \sqrt{11} (- \sqrt{2}) - \sqrt{2} \sqrt{11} - \sqrt{2} (- \sqrt{2})}{9}$

Step 8

Simplify and combine like terms.

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Step 8.1

Simplify each term.

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Step 8.1.1

Combine using the product rule for radicals.

$\frac{\sqrt{11 \cdot 11} + \sqrt{11} (- \sqrt{2}) - \sqrt{2} \sqrt{11} - \sqrt{2} (- \sqrt{2})}{9}$

Step 8.1.2

Multiply $11$ by $11$ .

$\frac{\sqrt{121} + \sqrt{11} (- \sqrt{2}) - \sqrt{2} \sqrt{11} - \sqrt{2} (- \sqrt{2})}{9}$

Step 8.1.3

Rewrite $121$ as $11^{2}$ .

$\frac{\sqrt{11^{2}} + \sqrt{11} (- \sqrt{2}) - \sqrt{2} \sqrt{11} - \sqrt{2} (- \sqrt{2})}{9}$

Step 8.1.4

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

$\frac{11 + \sqrt{11} (- \sqrt{2}) - \sqrt{2} \sqrt{11} - \sqrt{2} (- \sqrt{2})}{9}$

Step 8.1.5

Multiply $\sqrt{11} (- \sqrt{2})$ .

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Step 8.1.5.1

Combine using the product rule for radicals.

$\frac{11 - \sqrt{11 \cdot 2} - \sqrt{2} \sqrt{11} - \sqrt{2} (- \sqrt{2})}{9}$

Step 8.1.5.2

Multiply $11$ by $2$ .

$\frac{11 - \sqrt{22} - \sqrt{2} \sqrt{11} - \sqrt{2} (- \sqrt{2})}{9}$

Step 8.1.6

Multiply $- \sqrt{2} \sqrt{11}$ .

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Step 8.1.6.1

Combine using the product rule for radicals.

$\frac{11 - \sqrt{22} - \sqrt{11 \cdot 2} - \sqrt{2} (- \sqrt{2})}{9}$

Step 8.1.6.2

Multiply $11$ by $2$ .

$\frac{11 - \sqrt{22} - \sqrt{22} - \sqrt{2} (- \sqrt{2})}{9}$

Step 8.1.7

Multiply $- \sqrt{2} (- \sqrt{2})$ .

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Step 8.1.7.1

Multiply $- 1$ by $- 1$ .

$\frac{11 - \sqrt{22} - \sqrt{22} + 1 \sqrt{2} \sqrt{2}}{9}$

Step 8.1.7.2

Multiply $\sqrt{2}$ by $1$ .

$\frac{11 - \sqrt{22} - \sqrt{22} + \sqrt{2} \sqrt{2}}{9}$

Step 8.1.7.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{11 - \sqrt{22} - \sqrt{22} + {\sqrt{2}}^{1} \sqrt{2}}{9}$

Step 8.1.7.4

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{11 - \sqrt{22} - \sqrt{22} + {\sqrt{2}}^{1} {\sqrt{2}}^{1}}{9}$

Step 8.1.7.5

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

$\frac{11 - \sqrt{22} - \sqrt{22} + {\sqrt{2}}^{1 + 1}}{9}$

Step 8.1.7.6

Add $1$ and $1$ .

$\frac{11 - \sqrt{22} - \sqrt{22} + {\sqrt{2}}^{2}}{9}$

Step 8.1.8

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

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Step 8.1.8.1

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

$\frac{11 - \sqrt{22} - \sqrt{22} + {(2^{\frac{1}{2}})}^{2}}{9}$

Step 8.1.8.2

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

$\frac{11 - \sqrt{22} - \sqrt{22} + 2^{\frac{1}{2} \cdot 2}}{9}$

Step 8.1.8.3

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

$\frac{11 - \sqrt{22} - \sqrt{22} + 2^{\frac{2}{2}}}{9}$

Step 8.1.8.4

Cancel the common factor of $2$ .

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Step 8.1.8.4.1

Cancel the common factor.

$\frac{11 - \sqrt{22} - \sqrt{22} + 2^{\frac{2}{2}}}{9}$

Step 8.1.8.4.2

Rewrite the expression.

$\frac{11 - \sqrt{22} - \sqrt{22} + 2^{1}}{9}$

Step 8.1.8.5

Evaluate the exponent.

$\frac{11 - \sqrt{22} - \sqrt{22} + 2}{9}$

Step 8.2

Add $11$ and $2$ .

$\frac{13 - \sqrt{22} - \sqrt{22}}{9}$

Step 8.3

Subtract $\sqrt{22}$ from $- \sqrt{22}$ .

$\frac{13 - 2 \sqrt{22}}{9}$

Step 9

Split the fraction $\frac{13 - 2 \sqrt{22}}{9}$ into two fractions.

$\frac{13}{9} + \frac{- 2 \sqrt{22}}{9}$

Step 10

Move the negative in front of the fraction.

$\frac{13}{9} - \frac{2 \sqrt{22}}{9}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{13}{9} - \frac{2 \sqrt{22}}{9}$

Decimal Form:

$0.40212983 \dots$