Algebra Examples

$\frac{\sqrt{11} - \sqrt{3}}{\sqrt{11} + \sqrt{3}}$

Step 1

Multiply $\frac{\sqrt{11} - \sqrt{3}}{\sqrt{11} + \sqrt{3}}$ by $\frac{\sqrt{11} - \sqrt{3}}{\sqrt{11} - \sqrt{3}}$ .

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

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{\sqrt{11} - \sqrt{3}}{\sqrt{11} + \sqrt{3}}$ by $\frac{\sqrt{11} - \sqrt{3}}{\sqrt{11} - \sqrt{3}}$ .

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

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(\sqrt{11} - \sqrt{3}) (\sqrt{11} - \sqrt{3})}{{\sqrt{11}}^{2} - \sqrt{33} + \sqrt{33} - {\sqrt{3}}^{2}}$

Step 2.3

Simplify.

$\frac{(\sqrt{11} - \sqrt{3}) (\sqrt{11} - \sqrt{3})}{8}$

Step 3

Simplify the numerator.

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Step 3.1

Raise $\sqrt{11} - \sqrt{3}$ to the power of $1$ .

$\frac{{(\sqrt{11} - \sqrt{3})}^{1} (\sqrt{11} - \sqrt{3})}{8}$

Step 3.2

Raise $\sqrt{11} - \sqrt{3}$ to the power of $1$ .

$\frac{{(\sqrt{11} - \sqrt{3})}^{1} {(\sqrt{11} - \sqrt{3})}^{1}}{8}$

Step 3.3

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

$\frac{{(\sqrt{11} - \sqrt{3})}^{1 + 1}}{8}$

Step 3.4

Add $1$ and $1$ .

$\frac{{(\sqrt{11} - \sqrt{3})}^{2}}{8}$

Step 4

Rewrite ${(\sqrt{11} - \sqrt{3})}^{2}$ as $(\sqrt{11} - \sqrt{3}) (\sqrt{11} - \sqrt{3})$ .

$\frac{(\sqrt{11} - \sqrt{3}) (\sqrt{11} - \sqrt{3})}{8}$

Step 5

Expand $(\sqrt{11} - \sqrt{3}) (\sqrt{11} - \sqrt{3})$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$\frac{\sqrt{11} (\sqrt{11} - \sqrt{3}) - \sqrt{3} (\sqrt{11} - \sqrt{3})}{8}$

Step 5.2

Apply the distributive property.

$\frac{\sqrt{11} \sqrt{11} + \sqrt{11} (- \sqrt{3}) - \sqrt{3} (\sqrt{11} - \sqrt{3})}{8}$

Step 5.3

Apply the distributive property.

$\frac{\sqrt{11} \sqrt{11} + \sqrt{11} (- \sqrt{3}) - \sqrt{3} \sqrt{11} - \sqrt{3} (- \sqrt{3})}{8}$

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.

$\frac{\sqrt{11 \cdot 11} + \sqrt{11} (- \sqrt{3}) - \sqrt{3} \sqrt{11} - \sqrt{3} (- \sqrt{3})}{8}$

Step 6.1.2

Multiply $11$ by $11$ .

$\frac{\sqrt{121} + \sqrt{11} (- \sqrt{3}) - \sqrt{3} \sqrt{11} - \sqrt{3} (- \sqrt{3})}{8}$

Step 6.1.3

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

$\frac{\sqrt{11^{2}} + \sqrt{11} (- \sqrt{3}) - \sqrt{3} \sqrt{11} - \sqrt{3} (- \sqrt{3})}{8}$

Step 6.1.4

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

$\frac{11 + \sqrt{11} (- \sqrt{3}) - \sqrt{3} \sqrt{11} - \sqrt{3} (- \sqrt{3})}{8}$

Step 6.1.5

Multiply $\sqrt{11} (- \sqrt{3})$ .

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Step 6.1.5.1

Combine using the product rule for radicals.

$\frac{11 - \sqrt{11 \cdot 3} - \sqrt{3} \sqrt{11} - \sqrt{3} (- \sqrt{3})}{8}$

Step 6.1.5.2

Multiply $11$ by $3$ .

$\frac{11 - \sqrt{33} - \sqrt{3} \sqrt{11} - \sqrt{3} (- \sqrt{3})}{8}$

Step 6.1.6

Multiply $- \sqrt{3} \sqrt{11}$ .

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Step 6.1.6.1

Combine using the product rule for radicals.

$\frac{11 - \sqrt{33} - \sqrt{11 \cdot 3} - \sqrt{3} (- \sqrt{3})}{8}$

Step 6.1.6.2

Multiply $11$ by $3$ .

$\frac{11 - \sqrt{33} - \sqrt{33} - \sqrt{3} (- \sqrt{3})}{8}$

Step 6.1.7

Multiply $- \sqrt{3} (- \sqrt{3})$ .

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Step 6.1.7.1

Multiply $- 1$ by $- 1$ .

$\frac{11 - \sqrt{33} - \sqrt{33} + 1 \sqrt{3} \sqrt{3}}{8}$

Step 6.1.7.2

Multiply $\sqrt{3}$ by $1$ .

$\frac{11 - \sqrt{33} - \sqrt{33} + \sqrt{3} \sqrt{3}}{8}$

Step 6.1.7.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{11 - \sqrt{33} - \sqrt{33} + {\sqrt{3}}^{1} \sqrt{3}}{8}$

Step 6.1.7.4

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{11 - \sqrt{33} - \sqrt{33} + {\sqrt{3}}^{1} {\sqrt{3}}^{1}}{8}$

Step 6.1.7.5

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

$\frac{11 - \sqrt{33} - \sqrt{33} + {\sqrt{3}}^{1 + 1}}{8}$

Step 6.1.7.6

Add $1$ and $1$ .

$\frac{11 - \sqrt{33} - \sqrt{33} + {\sqrt{3}}^{2}}{8}$

Step 6.1.8

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 6.1.8.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{11 - \sqrt{33} - \sqrt{33} + {(3^{\frac{1}{2}})}^{2}}{8}$

Step 6.1.8.2

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

$\frac{11 - \sqrt{33} - \sqrt{33} + 3^{\frac{1}{2} \cdot 2}}{8}$

Step 6.1.8.3

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

$\frac{11 - \sqrt{33} - \sqrt{33} + 3^{\frac{2}{2}}}{8}$

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.

$\frac{11 - \sqrt{33} - \sqrt{33} + 3^{\frac{2}{2}}}{8}$

Step 6.1.8.4.2

Rewrite the expression.

$\frac{11 - \sqrt{33} - \sqrt{33} + 3^{1}}{8}$

Step 6.1.8.5

Evaluate the exponent.

$\frac{11 - \sqrt{33} - \sqrt{33} + 3}{8}$

Step 6.2

Add $11$ and $3$ .

$\frac{14 - \sqrt{33} - \sqrt{33}}{8}$

Step 6.3

Subtract $\sqrt{33}$ from $- \sqrt{33}$ .

$\frac{14 - 2 \sqrt{33}}{8}$

Step 7

Cancel the common factor of $14 - 2 \sqrt{33}$ and $8$ .

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Step 7.1

Factor $2$ out of $14$ .

$\frac{2 (7) - 2 \sqrt{33}}{8}$

Step 7.2

Factor $2$ out of $- 2 \sqrt{33}$ .

$\frac{2 (7) + 2 (- \sqrt{33})}{8}$

Step 7.3

Factor $2$ out of $2 (7) + 2 (- \sqrt{33})$ .

$\frac{2 (7 - \sqrt{33})}{8}$

Step 7.4

Cancel the common factors.

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Step 7.4.1

Factor $2$ out of $8$ .

$\frac{2 (7 - \sqrt{33})}{2 \cdot 4}$

Step 7.4.2

Cancel the common factor.

$\frac{2 (7 - \sqrt{33})}{2 \cdot 4}$

Step 7.4.3

Rewrite the expression.

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.31385933 \dots$