Evaluate ( square root of 7- square root of 8)/( square root of 7+ square root of 8)

$\frac{\sqrt{7} - \sqrt{8}}{\sqrt{7} + \sqrt{8}}$

Step 1

Simplify the numerator.

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

$\frac{\sqrt{7} - \sqrt{4 (2)}}{\sqrt{7} + \sqrt{8}}$

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{7} - \sqrt{2^{2} \cdot 2}}{\sqrt{7} + \sqrt{8}}$

Pull terms out from under the radical.

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

Multiply $2$ by $- 1$ .

$\frac{\sqrt{7} - 2 \sqrt{2}}{\sqrt{7} + \sqrt{8}}$

Step 2

Simplify the denominator.

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

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

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{7} - 2 \sqrt{2}}{\sqrt{7} + \sqrt{2^{2} \cdot 2}}$

Pull terms out from under the radical.

$\frac{\sqrt{7} - 2 \sqrt{2}}{\sqrt{7} + 2 \sqrt{2}}$

Step 3

Multiply $\frac{\sqrt{7} - 2 \sqrt{2}}{\sqrt{7} + 2 \sqrt{2}}$ by $\frac{\sqrt{7} - 2 \sqrt{2}}{\sqrt{7} - 2 \sqrt{2}}$ .

$\frac{\sqrt{7} - 2 \sqrt{2}}{\sqrt{7} + 2 \sqrt{2}} \cdot \frac{\sqrt{7} - 2 \sqrt{2}}{\sqrt{7} - 2 \sqrt{2}}$

Step 4

Combine fractions.

Multiply $\frac{\sqrt{7} - 2 \sqrt{2}}{\sqrt{7} + 2 \sqrt{2}}$ by $\frac{\sqrt{7} - 2 \sqrt{2}}{\sqrt{7} - 2 \sqrt{2}}$ .

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

Expand the denominator using the FOIL method.

$\frac{(\sqrt{7} - 2 \sqrt{2}) (\sqrt{7} - 2 \sqrt{2})}{{\sqrt{7}}^{2} - 2 \sqrt{14} + 2 \sqrt{14} - 4 {\sqrt{2}}^{2}}$

Simplify.

$\frac{(\sqrt{7} - 2 \sqrt{2}) (\sqrt{7} - 2 \sqrt{2})}{- 1}$

Simplify the expression.

Move the negative one from the denominator of $\frac{(\sqrt{7} - 2 \sqrt{2}) (\sqrt{7} - 2 \sqrt{2})}{- 1}$ .

$- 1 \cdot ((\sqrt{7} - 2 \sqrt{2}) (\sqrt{7} - 2 \sqrt{2}))$

Rewrite $- 1 \cdot ((\sqrt{7} - 2 \sqrt{2}) (\sqrt{7} - 2 \sqrt{2}))$ as $- ((\sqrt{7} - 2 \sqrt{2}) (\sqrt{7} - 2 \sqrt{2}))$ .

$- ((\sqrt{7} - 2 \sqrt{2}) (\sqrt{7} - 2 \sqrt{2}))$

Step 5

Expand $(\sqrt{7} - 2 \sqrt{2}) (\sqrt{7} - 2 \sqrt{2})$ using the FOIL Method.

Apply the distributive property.

$- (\sqrt{7} (\sqrt{7} - 2 \sqrt{2}) - 2 \sqrt{2} (\sqrt{7} - 2 \sqrt{2}))$

Apply the distributive property.

$- (\sqrt{7} \sqrt{7} + \sqrt{7} (- 2 \sqrt{2}) - 2 \sqrt{2} (\sqrt{7} - 2 \sqrt{2}))$

Apply the distributive property.

$- (\sqrt{7} \sqrt{7} + \sqrt{7} (- 2 \sqrt{2}) - 2 \sqrt{2} \sqrt{7} - 2 \sqrt{2} (- 2 \sqrt{2}))$

Step 6

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

$- (\sqrt{7 \cdot 7} + \sqrt{7} (- 2 \sqrt{2}) - 2 \sqrt{2} \sqrt{7} - 2 \sqrt{2} (- 2 \sqrt{2}))$

Multiply $7$ by $7$ .

$- (\sqrt{49} + \sqrt{7} (- 2 \sqrt{2}) - 2 \sqrt{2} \sqrt{7} - 2 \sqrt{2} (- 2 \sqrt{2}))$

Rewrite $49$ as $7^{2}$ .

$- (\sqrt{7^{2}} + \sqrt{7} (- 2 \sqrt{2}) - 2 \sqrt{2} \sqrt{7} - 2 \sqrt{2} (- 2 \sqrt{2}))$

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

$- (7 + \sqrt{7} (- 2 \sqrt{2}) - 2 \sqrt{2} \sqrt{7} - 2 \sqrt{2} (- 2 \sqrt{2}))$

Multiply $\sqrt{7} (- 2 \sqrt{2})$ .

Combine using the product rule for radicals.

$- (7 - 2 \sqrt{7 \cdot 2} - 2 \sqrt{2} \sqrt{7} - 2 \sqrt{2} (- 2 \sqrt{2}))$

Multiply $7$ by $2$ .

$- (7 - 2 \sqrt{14} - 2 \sqrt{2} \sqrt{7} - 2 \sqrt{2} (- 2 \sqrt{2}))$

Multiply $- 2 \sqrt{2} \sqrt{7}$ .

Combine using the product rule for radicals.

$- (7 - 2 \sqrt{14} - 2 \sqrt{7 \cdot 2} - 2 \sqrt{2} (- 2 \sqrt{2}))$

Multiply $7$ by $2$ .

$- (7 - 2 \sqrt{14} - 2 \sqrt{14} - 2 \sqrt{2} (- 2 \sqrt{2}))$

Multiply $- 2 \sqrt{2} (- 2 \sqrt{2})$ .

Multiply $- 2$ by $- 2$ .

$- (7 - 2 \sqrt{14} - 2 \sqrt{14} + 4 \sqrt{2} \sqrt{2})$

Raise $\sqrt{2}$ to the power of $1$ .

$- (7 - 2 \sqrt{14} - 2 \sqrt{14} + 4 ({\sqrt{2}}^{1} \sqrt{2}))$

Raise $\sqrt{2}$ to the power of $1$ .

$- (7 - 2 \sqrt{14} - 2 \sqrt{14} + 4 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}))$

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

$- (7 - 2 \sqrt{14} - 2 \sqrt{14} + 4 {\sqrt{2}}^{1 + 1})$

Add $1$ and $1$ .

$- (7 - 2 \sqrt{14} - 2 \sqrt{14} + 4 {\sqrt{2}}^{2})$

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

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

$- (7 - 2 \sqrt{14} - 2 \sqrt{14} + 4 {(2^{\frac{1}{2}})}^{2})$

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

$- (7 - 2 \sqrt{14} - 2 \sqrt{14} + 4 \cdot 2^{\frac{1}{2} \cdot 2})$

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

$- (7 - 2 \sqrt{14} - 2 \sqrt{14} + 4 \cdot 2^{\frac{2}{2}})$

Cancel the common factor of $2$ .

Cancel the common factor.

$- (7 - 2 \sqrt{14} - 2 \sqrt{14} + 4 \cdot 2^{\frac{2}{2}})$

Rewrite the expression.

$- (7 - 2 \sqrt{14} - 2 \sqrt{14} + 4 \cdot 2^{1})$

Evaluate the exponent.

$- (7 - 2 \sqrt{14} - 2 \sqrt{14} + 4 \cdot 2)$

Multiply $4$ by $2$ .

$- (7 - 2 \sqrt{14} - 2 \sqrt{14} + 8)$

Add $7$ and $8$ .

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

Subtract $2 \sqrt{14}$ from $- 2 \sqrt{14}$ .

$- (15 - 4 \sqrt{14})$

Step 7

Apply the distributive property.

$- 1 \cdot 15 - (- 4 \sqrt{14})$

Step 8

Multiply.

Multiply $- 1$ by $15$ .

$- 15 - (- 4 \sqrt{14})$

Multiply $- 4$ by $- 1$ .

$- 15 + 4 \sqrt{14}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- 15 + 4 \sqrt{14}$

Decimal Form:

$- 0.03337045 \dots$

Evaluate ( square root of 7- square root of 8)/( square root of 7+ square root of 8)

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