Evaluate ( square root of 14- square root of 15)/( square root of 14+ square root of 15)

$\frac{\sqrt{14} - \sqrt{15}}{\sqrt{14} + \sqrt{15}}$

Step 1

Multiply $\frac{\sqrt{14} - \sqrt{15}}{\sqrt{14} + \sqrt{15}}$ by $\frac{\sqrt{14} - \sqrt{15}}{\sqrt{14} - \sqrt{15}}$ .

$\frac{\sqrt{14} - \sqrt{15}}{\sqrt{14} + \sqrt{15}} \cdot \frac{\sqrt{14} - \sqrt{15}}{\sqrt{14} - \sqrt{15}}$

Step 2

Combine fractions.

Multiply $\frac{\sqrt{14} - \sqrt{15}}{\sqrt{14} + \sqrt{15}}$ by $\frac{\sqrt{14} - \sqrt{15}}{\sqrt{14} - \sqrt{15}}$ .

$\frac{(\sqrt{14} - \sqrt{15}) (\sqrt{14} - \sqrt{15})}{(\sqrt{14} + \sqrt{15}) (\sqrt{14} - \sqrt{15})}$

Expand the denominator using the FOIL method.

$\frac{(\sqrt{14} - \sqrt{15}) (\sqrt{14} - \sqrt{15})}{{\sqrt{14}}^{2} - \sqrt{210} + \sqrt{210} - {\sqrt{15}}^{2}}$

Simplify.

$\frac{(\sqrt{14} - \sqrt{15}) (\sqrt{14} - \sqrt{15})}{- 1}$

Simplify the expression.

Move the negative one from the denominator of $\frac{(\sqrt{14} - \sqrt{15}) (\sqrt{14} - \sqrt{15})}{- 1}$ .

$- 1 \cdot ((\sqrt{14} - \sqrt{15}) (\sqrt{14} - \sqrt{15}))$

Rewrite $- 1 \cdot ((\sqrt{14} - \sqrt{15}) (\sqrt{14} - \sqrt{15}))$ as $- ((\sqrt{14} - \sqrt{15}) (\sqrt{14} - \sqrt{15}))$ .

$- ((\sqrt{14} - \sqrt{15}) (\sqrt{14} - \sqrt{15}))$

Step 3

Expand $(\sqrt{14} - \sqrt{15}) (\sqrt{14} - \sqrt{15})$ using the FOIL Method.

Apply the distributive property.

$- (\sqrt{14} (\sqrt{14} - \sqrt{15}) - \sqrt{15} (\sqrt{14} - \sqrt{15}))$

Apply the distributive property.

$- (\sqrt{14} \sqrt{14} + \sqrt{14} (- \sqrt{15}) - \sqrt{15} (\sqrt{14} - \sqrt{15}))$

Apply the distributive property.

$- (\sqrt{14} \sqrt{14} + \sqrt{14} (- \sqrt{15}) - \sqrt{15} \sqrt{14} - \sqrt{15} (- \sqrt{15}))$

Step 4

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

$- (\sqrt{14 \cdot 14} + \sqrt{14} (- \sqrt{15}) - \sqrt{15} \sqrt{14} - \sqrt{15} (- \sqrt{15}))$

Multiply $14$ by $14$ .

$- (\sqrt{196} + \sqrt{14} (- \sqrt{15}) - \sqrt{15} \sqrt{14} - \sqrt{15} (- \sqrt{15}))$

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

$- (\sqrt{14^{2}} + \sqrt{14} (- \sqrt{15}) - \sqrt{15} \sqrt{14} - \sqrt{15} (- \sqrt{15}))$

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

$- (14 + \sqrt{14} (- \sqrt{15}) - \sqrt{15} \sqrt{14} - \sqrt{15} (- \sqrt{15}))$

Multiply $\sqrt{14} (- \sqrt{15})$ .

Combine using the product rule for radicals.

$- (14 - \sqrt{14 \cdot 15} - \sqrt{15} \sqrt{14} - \sqrt{15} (- \sqrt{15}))$

Multiply $14$ by $15$ .

$- (14 - \sqrt{210} - \sqrt{15} \sqrt{14} - \sqrt{15} (- \sqrt{15}))$

Multiply $- \sqrt{15} \sqrt{14}$ .

Combine using the product rule for radicals.

$- (14 - \sqrt{210} - \sqrt{14 \cdot 15} - \sqrt{15} (- \sqrt{15}))$

Multiply $14$ by $15$ .

$- (14 - \sqrt{210} - \sqrt{210} - \sqrt{15} (- \sqrt{15}))$

Multiply $- \sqrt{15} (- \sqrt{15})$ .

Multiply $- 1$ by $- 1$ .

$- (14 - \sqrt{210} - \sqrt{210} + 1 \sqrt{15} \sqrt{15})$

Multiply $\sqrt{15}$ by $1$ .

$- (14 - \sqrt{210} - \sqrt{210} + \sqrt{15} \sqrt{15})$

Raise $\sqrt{15}$ to the power of $1$ .

$- (14 - \sqrt{210} - \sqrt{210} + {\sqrt{15}}^{1} \sqrt{15})$

Raise $\sqrt{15}$ to the power of $1$ .

$- (14 - \sqrt{210} - \sqrt{210} + {\sqrt{15}}^{1} {\sqrt{15}}^{1})$

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

$- (14 - \sqrt{210} - \sqrt{210} + {\sqrt{15}}^{1 + 1})$

Add $1$ and $1$ .

$- (14 - \sqrt{210} - \sqrt{210} + {\sqrt{15}}^{2})$

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$- (14 - \sqrt{210} - \sqrt{210} + {(15^{\frac{1}{2}})}^{2})$

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

$- (14 - \sqrt{210} - \sqrt{210} + 15^{\frac{1}{2} \cdot 2})$

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

$- (14 - \sqrt{210} - \sqrt{210} + 15^{\frac{2}{2}})$

Cancel the common factor of $2$ .

Cancel the common factor.

$- (14 - \sqrt{210} - \sqrt{210} + 15^{\frac{2}{2}})$

Rewrite the expression.

$- (14 - \sqrt{210} - \sqrt{210} + 15^{1})$

Evaluate the exponent.

$- (14 - \sqrt{210} - \sqrt{210} + 15)$

Add $14$ and $15$ .

$- (29 - \sqrt{210} - \sqrt{210})$

Subtract $\sqrt{210}$ from $- \sqrt{210}$ .

$- (29 - 2 \sqrt{210})$

Step 5

Apply the distributive property.

$- 1 \cdot 29 - (- 2 \sqrt{210})$

Step 6

Multiply.

Multiply $- 1$ by $29$ .

$- 29 - (- 2 \sqrt{210})$

Multiply $- 2$ by $- 1$ .

$- 29 + 2 \sqrt{210}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- 29 + 2 \sqrt{210}$

Decimal Form:

$- 0.01724650 \dots$

Evaluate ( square root of 14- square root of 15)/( square root of 14+ square root of 15)

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