Algebra Examples

$\frac{\sqrt{11}}{\sqrt{13} + \sqrt{11}}$

Step 1

Multiply $\frac{\sqrt{11}}{\sqrt{13} + \sqrt{11}}$ by $\frac{\sqrt{13} - \sqrt{11}}{\sqrt{13} - \sqrt{11}}$ .

$\frac{\sqrt{11}}{\sqrt{13} + \sqrt{11}} \cdot \frac{\sqrt{13} - \sqrt{11}}{\sqrt{13} - \sqrt{11}}$

Step 2

Multiply $\frac{\sqrt{11}}{\sqrt{13} + \sqrt{11}}$ by $\frac{\sqrt{13} - \sqrt{11}}{\sqrt{13} - \sqrt{11}}$ .

$\frac{\sqrt{11} (\sqrt{13} - \sqrt{11})}{(\sqrt{13} + \sqrt{11}) (\sqrt{13} - \sqrt{11})}$

Step 3

Expand the denominator using the FOIL method.

$\frac{\sqrt{11} (\sqrt{13} - \sqrt{11})}{{\sqrt{13}}^{2} - \sqrt{143} + \sqrt{143} - {\sqrt{11}}^{2}}$

Step 4

Simplify.

$\frac{\sqrt{11} (\sqrt{13} - \sqrt{11})}{2}$

Step 5

Apply the distributive property.

$\frac{\sqrt{11} \sqrt{13} + \sqrt{11} (- \sqrt{11})}{2}$

Step 6

Combine using the product rule for radicals.

$\frac{\sqrt{11 \cdot 13} + \sqrt{11} (- \sqrt{11})}{2}$

Step 7

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

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Step 7.1

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

$\frac{\sqrt{11 \cdot 13} - ({\sqrt{11}}^{1} \sqrt{11})}{2}$

Step 7.2

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

$\frac{\sqrt{11 \cdot 13} - ({\sqrt{11}}^{1} {\sqrt{11}}^{1})}{2}$

Step 7.3

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

$\frac{\sqrt{11 \cdot 13} - {\sqrt{11}}^{1 + 1}}{2}$

Step 7.4

Add $1$ and $1$ .

$\frac{\sqrt{11 \cdot 13} - {\sqrt{11}}^{2}}{2}$

Step 8

Simplify each term.

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Step 8.1

Multiply $11$ by $13$ .

$\frac{\sqrt{143} - {\sqrt{11}}^{2}}{2}$

Step 8.2

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

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Step 8.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$\frac{\sqrt{143} - {(11^{\frac{1}{2}})}^{2}}{2}$

Step 8.2.2

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

$\frac{\sqrt{143} - 11^{\frac{1}{2} \cdot 2}}{2}$

Step 8.2.3

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

$\frac{\sqrt{143} - 11^{\frac{2}{2}}}{2}$

Step 8.2.4

Cancel the common factor of $2$ .

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Step 8.2.4.1

Cancel the common factor.

$\frac{\sqrt{143} - 11^{\frac{2}{2}}}{2}$

Step 8.2.4.2

Rewrite the expression.

$\frac{\sqrt{143} - 11^{1}}{2}$

Step 8.2.5

Evaluate the exponent.

$\frac{\sqrt{143} - 1 \cdot 11}{2}$

Step 8.3

Multiply $- 1$ by $11$ .

$\frac{\sqrt{143} - 11}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{143} - 11}{2}$

Decimal Form:

$0.47913037 \dots$