Evaluate ( square root of 14- square root of 13)/( square root of 14+ square root of 13)

$\frac{\sqrt{14} - \sqrt{13}}{\sqrt{14} + \sqrt{13}}$

Step 1

Multiply $\frac{\sqrt{14} - \sqrt{13}}{\sqrt{14} + \sqrt{13}}$ by $\frac{\sqrt{14} - \sqrt{13}}{\sqrt{14} - \sqrt{13}}$ .

$\frac{\sqrt{14} - \sqrt{13}}{\sqrt{14} + \sqrt{13}} \cdot \frac{\sqrt{14} - \sqrt{13}}{\sqrt{14} - \sqrt{13}}$

Step 2

Combine fractions.

Multiply $\frac{\sqrt{14} - \sqrt{13}}{\sqrt{14} + \sqrt{13}}$ by $\frac{\sqrt{14} - \sqrt{13}}{\sqrt{14} - \sqrt{13}}$ .

$\frac{(\sqrt{14} - \sqrt{13}) (\sqrt{14} - \sqrt{13})}{(\sqrt{14} + \sqrt{13}) (\sqrt{14} - \sqrt{13})}$

Expand the denominator using the FOIL method.

$\frac{(\sqrt{14} - \sqrt{13}) (\sqrt{14} - \sqrt{13})}{{\sqrt{14}}^{2} - \sqrt{182} + \sqrt{182} - {\sqrt{13}}^{2}}$

Simplify.

$\frac{(\sqrt{14} - \sqrt{13}) (\sqrt{14} - \sqrt{13})}{1}$

Divide $(\sqrt{14} - \sqrt{13}) (\sqrt{14} - \sqrt{13})$ by $1$ .

$(\sqrt{14} - \sqrt{13}) (\sqrt{14} - \sqrt{13})$

Step 3

Expand $(\sqrt{14} - \sqrt{13}) (\sqrt{14} - \sqrt{13})$ using the FOIL Method.

Apply the distributive property.

$\sqrt{14} (\sqrt{14} - \sqrt{13}) - \sqrt{13} (\sqrt{14} - \sqrt{13})$

Apply the distributive property.

$\sqrt{14} \sqrt{14} + \sqrt{14} (- \sqrt{13}) - \sqrt{13} (\sqrt{14} - \sqrt{13})$

Apply the distributive property.

$\sqrt{14} \sqrt{14} + \sqrt{14} (- \sqrt{13}) - \sqrt{13} \sqrt{14} - \sqrt{13} (- \sqrt{13})$

Step 4

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

$\sqrt{14 \cdot 14} + \sqrt{14} (- \sqrt{13}) - \sqrt{13} \sqrt{14} - \sqrt{13} (- \sqrt{13})$

Multiply $14$ by $14$ .

$\sqrt{196} + \sqrt{14} (- \sqrt{13}) - \sqrt{13} \sqrt{14} - \sqrt{13} (- \sqrt{13})$

Rewrite $196$ as $14^{2}$ .

$\sqrt{14^{2}} + \sqrt{14} (- \sqrt{13}) - \sqrt{13} \sqrt{14} - \sqrt{13} (- \sqrt{13})$

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

$14 + \sqrt{14} (- \sqrt{13}) - \sqrt{13} \sqrt{14} - \sqrt{13} (- \sqrt{13})$

Multiply $\sqrt{14} (- \sqrt{13})$ .

Combine using the product rule for radicals.

$14 - \sqrt{14 \cdot 13} - \sqrt{13} \sqrt{14} - \sqrt{13} (- \sqrt{13})$

Multiply $14$ by $13$ .

$14 - \sqrt{182} - \sqrt{13} \sqrt{14} - \sqrt{13} (- \sqrt{13})$

Multiply $- \sqrt{13} \sqrt{14}$ .

Combine using the product rule for radicals.

$14 - \sqrt{182} - \sqrt{14 \cdot 13} - \sqrt{13} (- \sqrt{13})$

Multiply $14$ by $13$ .

$14 - \sqrt{182} - \sqrt{182} - \sqrt{13} (- \sqrt{13})$

Multiply $- \sqrt{13} (- \sqrt{13})$ .

Multiply $- 1$ by $- 1$ .

$14 - \sqrt{182} - \sqrt{182} + 1 \sqrt{13} \sqrt{13}$

Multiply $\sqrt{13}$ by $1$ .

$14 - \sqrt{182} - \sqrt{182} + \sqrt{13} \sqrt{13}$

Raise $\sqrt{13}$ to the power of $1$ .

$14 - \sqrt{182} - \sqrt{182} + {\sqrt{13}}^{1} \sqrt{13}$

Raise $\sqrt{13}$ to the power of $1$ .

$14 - \sqrt{182} - \sqrt{182} + {\sqrt{13}}^{1} {\sqrt{13}}^{1}$

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

$14 - \sqrt{182} - \sqrt{182} + {\sqrt{13}}^{1 + 1}$

Add $1$ and $1$ .

$14 - \sqrt{182} - \sqrt{182} + {\sqrt{13}}^{2}$

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$14 - \sqrt{182} - \sqrt{182} + {(13^{\frac{1}{2}})}^{2}$

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

$14 - \sqrt{182} - \sqrt{182} + 13^{\frac{1}{2} \cdot 2}$

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

$14 - \sqrt{182} - \sqrt{182} + 13^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$14 - \sqrt{182} - \sqrt{182} + 13^{\frac{2}{2}}$

Rewrite the expression.

$14 - \sqrt{182} - \sqrt{182} + 13^{1}$

Evaluate the exponent.

$14 - \sqrt{182} - \sqrt{182} + 13$

Add $14$ and $13$ .

$27 - \sqrt{182} - \sqrt{182}$

Subtract $\sqrt{182}$ from $- \sqrt{182}$ .

$27 - 2 \sqrt{182}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$27 - 2 \sqrt{182}$

Decimal Form:

$0.01852487 \dots$

Evaluate ( square root of 14- square root of 13)/( square root of 14+ square root of 13)

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