Simplify ( square root of 82)/(9- square root of 82)

$\frac{\sqrt{82}}{9 - \sqrt{82}}$

Step 1

Multiply $\frac{\sqrt{82}}{9 - \sqrt{82}}$ by $\frac{9 + \sqrt{82}}{9 + \sqrt{82}}$ .

$\frac{\sqrt{82}}{9 - \sqrt{82}} \cdot \frac{9 + \sqrt{82}}{9 + \sqrt{82}}$

Step 2

Simplify terms.

Multiply $\frac{\sqrt{82}}{9 - \sqrt{82}}$ by $\frac{9 + \sqrt{82}}{9 + \sqrt{82}}$ .

$\frac{\sqrt{82} (9 + \sqrt{82})}{(9 - \sqrt{82}) (9 + \sqrt{82})}$

Expand the denominator using the FOIL method.

$\frac{\sqrt{82} (9 + \sqrt{82})}{81 + 9 \sqrt{82} - 9 \sqrt{82} - {\sqrt{82}}^{2}}$

Simplify.

$\frac{\sqrt{82} (9 + \sqrt{82})}{- 1}$

Simplify the expression.

Move the negative one from the denominator of $\frac{\sqrt{82} (9 + \sqrt{82})}{- 1}$ .

$- 1 \cdot (\sqrt{82} (9 + \sqrt{82}))$

Rewrite $- 1 \cdot (\sqrt{82} (9 + \sqrt{82}))$ as $- (\sqrt{82} (9 + \sqrt{82}))$ .

$- (\sqrt{82} (9 + \sqrt{82}))$

Apply the distributive property.

$- (\sqrt{82} \cdot 9 + \sqrt{82} \sqrt{82})$

Move $9$ to the left of $\sqrt{82}$ .

$- (9 \cdot \sqrt{82} + \sqrt{82} \sqrt{82})$

Combine using the product rule for radicals.

$- (9 \cdot \sqrt{82} + \sqrt{82 \cdot 82})$

Step 3

Simplify each term.

Multiply $82$ by $82$ .

$- (9 \sqrt{82} + \sqrt{6724})$

Rewrite $6724$ as $82^{2}$ .

$- (9 \sqrt{82} + \sqrt{82^{2}})$

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

$- (9 \sqrt{82} + 82)$

Step 4

Simplify by multiplying through.

Apply the distributive property.

$- (9 \sqrt{82}) - 1 \cdot 82$

Multiply.

Multiply $9$ by $- 1$ .

$- 9 \sqrt{82} - 1 \cdot 82$

Multiply $- 1$ by $82$ .

$- 9 \sqrt{82} - 82$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- 9 \sqrt{82} - 82$

Decimal Form:

$- 163.49846624 \dots$