Simplify ( square root of 7-5)/( square root of 7+ square root of 8)

$\frac{\sqrt{7} - 5}{\sqrt{7} + \sqrt{8}}$

Step 1

Simplify the denominator.

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

Factor $4$ out of $8$ .

$\frac{\sqrt{7} - 5}{\sqrt{7} + \sqrt{4 (2)}}$

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

$\frac{\sqrt{7} - 5}{\sqrt{7} + \sqrt{2^{2} \cdot 2}}$

Pull terms out from under the radical.

$\frac{\sqrt{7} - 5}{\sqrt{7} + 2 \sqrt{2}}$

Step 2

Multiply $\frac{\sqrt{7} - 5}{\sqrt{7} + 2 \sqrt{2}}$ by $\frac{\sqrt{7} - 2 \sqrt{2}}{\sqrt{7} - 2 \sqrt{2}}$ .

$\frac{\sqrt{7} - 5}{\sqrt{7} + 2 \sqrt{2}} \cdot \frac{\sqrt{7} - 2 \sqrt{2}}{\sqrt{7} - 2 \sqrt{2}}$

Step 3

Combine fractions.

Multiply $\frac{\sqrt{7} - 5}{\sqrt{7} + 2 \sqrt{2}}$ by $\frac{\sqrt{7} - 2 \sqrt{2}}{\sqrt{7} - 2 \sqrt{2}}$ .

$\frac{(\sqrt{7} - 5) (\sqrt{7} - 2 \sqrt{2})}{(\sqrt{7} + 2 \sqrt{2}) (\sqrt{7} - 2 \sqrt{2})}$

Expand the denominator using the FOIL method.

$\frac{(\sqrt{7} - 5) (\sqrt{7} - 2 \sqrt{2})}{{\sqrt{7}}^{2} - 2 \sqrt{14} + 2 \sqrt{14} - 4 {\sqrt{2}}^{2}}$

Simplify.

$\frac{(\sqrt{7} - 5) (\sqrt{7} - 2 \sqrt{2})}{- 1}$

Simplify the expression.

Move the negative one from the denominator of $\frac{(\sqrt{7} - 5) (\sqrt{7} - 2 \sqrt{2})}{- 1}$ .

$- 1 \cdot ((\sqrt{7} - 5) (\sqrt{7} - 2 \sqrt{2}))$

Rewrite $- 1 \cdot ((\sqrt{7} - 5) (\sqrt{7} - 2 \sqrt{2}))$ as $- ((\sqrt{7} - 5) (\sqrt{7} - 2 \sqrt{2}))$ .

$- ((\sqrt{7} - 5) (\sqrt{7} - 2 \sqrt{2}))$

Step 4

Expand $(\sqrt{7} - 5) (\sqrt{7} - 2 \sqrt{2})$ using the FOIL Method.

Apply the distributive property.

$- (\sqrt{7} (\sqrt{7} - 2 \sqrt{2}) - 5 (\sqrt{7} - 2 \sqrt{2}))$

Apply the distributive property.

$- (\sqrt{7} \sqrt{7} + \sqrt{7} (- 2 \sqrt{2}) - 5 (\sqrt{7} - 2 \sqrt{2}))$

Apply the distributive property.

$- (\sqrt{7} \sqrt{7} + \sqrt{7} (- 2 \sqrt{2}) - 5 \sqrt{7} - 5 (- 2 \sqrt{2}))$

Step 5

Simplify terms.

Simplify each term.

Combine using the product rule for radicals.

$- (\sqrt{7 \cdot 7} + \sqrt{7} (- 2 \sqrt{2}) - 5 \sqrt{7} - 5 (- 2 \sqrt{2}))$

Multiply $7$ by $7$ .

$- (\sqrt{49} + \sqrt{7} (- 2 \sqrt{2}) - 5 \sqrt{7} - 5 (- 2 \sqrt{2}))$

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

$- (\sqrt{7^{2}} + \sqrt{7} (- 2 \sqrt{2}) - 5 \sqrt{7} - 5 (- 2 \sqrt{2}))$

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

$- (7 + \sqrt{7} (- 2 \sqrt{2}) - 5 \sqrt{7} - 5 (- 2 \sqrt{2}))$

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

Combine using the product rule for radicals.

$- (7 - 2 \sqrt{7 \cdot 2} - 5 \sqrt{7} - 5 (- 2 \sqrt{2}))$

Multiply $7$ by $2$ .

$- (7 - 2 \sqrt{14} - 5 \sqrt{7} - 5 (- 2 \sqrt{2}))$

Multiply $- 2$ by $- 5$ .

$- (7 - 2 \sqrt{14} - 5 \sqrt{7} + 10 \sqrt{2})$

Apply the distributive property.

$- 1 \cdot 7 - (- 2 \sqrt{14}) - (- 5 \sqrt{7}) - (10 \sqrt{2})$

Step 6

Simplify.

Multiply $- 1$ by $7$ .

$- 7 - (- 2 \sqrt{14}) - (- 5 \sqrt{7}) - (10 \sqrt{2})$

Multiply $- 2$ by $- 1$ .

$- 7 + 2 \sqrt{14} - (- 5 \sqrt{7}) - (10 \sqrt{2})$

Multiply $- 5$ by $- 1$ .

$- 7 + 2 \sqrt{14} + 5 \sqrt{7} - (10 \sqrt{2})$

Multiply $10$ by $- 1$ .

$- 7 + 2 \sqrt{14} + 5 \sqrt{7} - 10 \sqrt{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- 7 + 2 \sqrt{14} + 5 \sqrt{7} - 10 \sqrt{2}$

Decimal Form:

$- 0.43006429 \dots$

Simplify ( square root of 7-5)/( square root of 7+ square root of 8)

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