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

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

Step 1

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

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

Step 2

Combine fractions.

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

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

Expand the denominator using the FOIL method.

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

Simplify.

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

Simplify the expression.

Move the negative one from the denominator of $\frac{(\sqrt{13} - \sqrt{14}) (\sqrt{13} - \sqrt{14})}{- 1}$ .

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

Rewrite $- 1 \cdot ((\sqrt{13} - \sqrt{14}) (\sqrt{13} - \sqrt{14}))$ as $- ((\sqrt{13} - \sqrt{14}) (\sqrt{13} - \sqrt{14}))$ .

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

Step 3

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 4

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

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

Multiply $13$ by $13$ .

$- (\sqrt{169} + \sqrt{13} (- \sqrt{14}) - \sqrt{14} \sqrt{13} - \sqrt{14} (- \sqrt{14}))$

Rewrite $169$ as $13^{2}$ .

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

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

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

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

Combine using the product rule for radicals.

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

Multiply $13$ by $14$ .

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

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

Combine using the product rule for radicals.

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

Multiply $13$ by $14$ .

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

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

Multiply $- 1$ by $- 1$ .

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

Multiply $\sqrt{14}$ by $1$ .

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

Raise $\sqrt{14}$ to the power of $1$ .

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

Raise $\sqrt{14}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{14}}^{2}$ as $14$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{14}$ as $14^{\frac{1}{2}}$ .

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Add $13$ and $14$ .

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

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

$- (27 - 2 \sqrt{182})$

Step 5

Apply the distributive property.

$- 1 \cdot 27 - (- 2 \sqrt{182})$

Step 6

Multiply.

Multiply $- 1$ by $27$ .

$- 27 - (- 2 \sqrt{182})$

Multiply $- 2$ by $- 1$ .

$- 27 + 2 \sqrt{182}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- 27 + 2 \sqrt{182}$

Decimal Form:

$- 0.01852487 \dots$

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

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