Evaluate (5- square root of 7)/(9- square root of 14)

$\frac{5 - \sqrt{7}}{9 - \sqrt{14}}$

Step 1

Multiply $\frac{5 - \sqrt{7}}{9 - \sqrt{14}}$ by $\frac{9 + \sqrt{14}}{9 + \sqrt{14}}$ .

$\frac{5 - \sqrt{7}}{9 - \sqrt{14}} \cdot \frac{9 + \sqrt{14}}{9 + \sqrt{14}}$

Step 2

Combine fractions.

Multiply $\frac{5 - \sqrt{7}}{9 - \sqrt{14}}$ by $\frac{9 + \sqrt{14}}{9 + \sqrt{14}}$ .

$\frac{(5 - \sqrt{7}) (9 + \sqrt{14})}{(9 - \sqrt{14}) (9 + \sqrt{14})}$

Expand the denominator using the FOIL method.

$\frac{(5 - \sqrt{7}) (9 + \sqrt{14})}{81 + 9 \sqrt{14} - 9 \sqrt{14} - {\sqrt{14}}^{2}}$

Simplify.

$\frac{(5 - \sqrt{7}) (9 + \sqrt{14})}{67}$

Step 3

Expand $(5 - \sqrt{7}) (9 + \sqrt{14})$ using the FOIL Method.

Apply the distributive property.

$\frac{5 (9 + \sqrt{14}) - \sqrt{7} (9 + \sqrt{14})}{67}$

Apply the distributive property.

$\frac{5 \cdot 9 + 5 \sqrt{14} - \sqrt{7} (9 + \sqrt{14})}{67}$

Apply the distributive property.

$\frac{5 \cdot 9 + 5 \sqrt{14} - \sqrt{7} \cdot 9 - \sqrt{7} \sqrt{14}}{67}$

Step 4

Simplify each term.

Multiply $5$ by $9$ .

$\frac{45 + 5 \sqrt{14} - \sqrt{7} \cdot 9 - \sqrt{7} \sqrt{14}}{67}$

Multiply $9$ by $- 1$ .

$\frac{45 + 5 \sqrt{14} - 9 \sqrt{7} - \sqrt{7} \sqrt{14}}{67}$

Multiply $- \sqrt{7} \sqrt{14}$ .

Combine using the product rule for radicals.

$\frac{45 + 5 \sqrt{14} - 9 \sqrt{7} - \sqrt{14 \cdot 7}}{67}$

Multiply $14$ by $7$ .

$\frac{45 + 5 \sqrt{14} - 9 \sqrt{7} - \sqrt{98}}{67}$

Rewrite $98$ as $7^{2} \cdot 2$ .

Factor $49$ out of $98$ .

$\frac{45 + 5 \sqrt{14} - 9 \sqrt{7} - \sqrt{49 (2)}}{67}$

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

$\frac{45 + 5 \sqrt{14} - 9 \sqrt{7} - \sqrt{7^{2} \cdot 2}}{67}$

Pull terms out from under the radical.

$\frac{45 + 5 \sqrt{14} - 9 \sqrt{7} - (7 \sqrt{2})}{67}$

Multiply $7$ by $- 1$ .

$\frac{45 + 5 \sqrt{14} - 9 \sqrt{7} - 7 \sqrt{2}}{67}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{45 + 5 \sqrt{14} - 9 \sqrt{7} - 7 \sqrt{2}}{67}$

Decimal Form:

$0.44771686 \dots$

Evaluate (5- square root of 7)/(9- square root of 14)

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