Simplify (9+ square root of 2)/(4- square root of 7)

$\frac{9 + \sqrt{2}}{4 - \sqrt{7}}$

Step 1

Multiply $\frac{9 + \sqrt{2}}{4 - \sqrt{7}}$ by $\frac{4 + \sqrt{7}}{4 + \sqrt{7}}$ .

$\frac{9 + \sqrt{2}}{4 - \sqrt{7}} \cdot \frac{4 + \sqrt{7}}{4 + \sqrt{7}}$

Step 2

Combine fractions.

Multiply $\frac{9 + \sqrt{2}}{4 - \sqrt{7}}$ by $\frac{4 + \sqrt{7}}{4 + \sqrt{7}}$ .

$\frac{(9 + \sqrt{2}) (4 + \sqrt{7})}{(4 - \sqrt{7}) (4 + \sqrt{7})}$

Expand the denominator using the FOIL method.

$\frac{(9 + \sqrt{2}) (4 + \sqrt{7})}{16 + 4 \sqrt{7} - 4 \sqrt{7} - {\sqrt{7}}^{2}}$

Simplify.

$\frac{(9 + \sqrt{2}) (4 + \sqrt{7})}{9}$

Step 3

Expand $(9 + \sqrt{2}) (4 + \sqrt{7})$ using the FOIL Method.

Apply the distributive property.

$\frac{9 (4 + \sqrt{7}) + \sqrt{2} (4 + \sqrt{7})}{9}$

Apply the distributive property.

$\frac{9 \cdot 4 + 9 \sqrt{7} + \sqrt{2} (4 + \sqrt{7})}{9}$

Apply the distributive property.

$\frac{9 \cdot 4 + 9 \sqrt{7} + \sqrt{2} \cdot 4 + \sqrt{2} \sqrt{7}}{9}$

Step 4

Simplify each term.

Multiply $9$ by $4$ .

$\frac{36 + 9 \sqrt{7} + \sqrt{2} \cdot 4 + \sqrt{2} \sqrt{7}}{9}$

Move $4$ to the left of $\sqrt{2}$ .

$\frac{36 + 9 \sqrt{7} + 4 \cdot \sqrt{2} + \sqrt{2} \sqrt{7}}{9}$

Combine using the product rule for radicals.

$\frac{36 + 9 \sqrt{7} + 4 \sqrt{2} + \sqrt{2 \cdot 7}}{9}$

Multiply $2$ by $7$ .

$\frac{36 + 9 \sqrt{7} + 4 \sqrt{2} + \sqrt{14}}{9}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{36 + 9 \sqrt{7} + 4 \sqrt{2} + \sqrt{14}}{9}$

Decimal Form:

$7.69003038 \dots$