Basic Math Examples

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

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

Step 1

Multiply

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

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

\frac{9 + \sqrt{82}}{9 + \sqrt{82}}

$\frac{9 + \sqrt{82}}{9 + \sqrt{82}}$ .

\frac{\sqrt{82}}{9 - \sqrt{82}} \cdot \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.

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Step 2.1

Multiply

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

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

\frac{9 + \sqrt{82}}{9 + \sqrt{82}}

$\frac{9 + \sqrt{82}}{9 + \sqrt{82}}$ .

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

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

Step 2.2

Expand the denominator using the FOIL method.

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

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

Step 2.3

Simplify.

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

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

Step 2.4

Simplify the expression.

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Step 2.4.1

Move the negative one from the denominator of

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

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

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

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

Step 2.4.2

Rewrite

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

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

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

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

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

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

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

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

Step 2.5

Apply the distributive property.

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

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

Step 2.6

Move

9

$9$ to the left of

\sqrt{82}

$\sqrt{82}$ .

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

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

Step 2.7

Combine using the product rule for radicals.

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

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

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

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

Step 3

Simplify each term.

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Step 3.1

Multiply

82

$82$ by

82

$82$ .

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

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

Step 3.2

Rewrite

6724

$6724$ as

82^{2}

$82^{2}$ .

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

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

Step 3.3

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

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

Step 4

Simplify by multiplying through.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply.

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Step 4.2.1

Multiply

$9$ by

$- 1$ .

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

Step 4.2.2

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:

$- 163.49846624 \dots$