Trigonometry Examples

$\sqrt{\frac{81}{82}}$

Step 1

Rewrite $\sqrt{\frac{81}{82}}$ as $\frac{\sqrt{81}}{\sqrt{82}}$ .

$\frac{\sqrt{81}}{\sqrt{82}}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $81$ as $9^{2}$ .

$\frac{\sqrt{9^{2}}}{\sqrt{82}}$

Step 2.2

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

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

Step 3

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

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

Step 4

Combine and simplify the denominator.

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Step 4.1

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

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

Step 4.2

Raise $\sqrt{82}$ to the power of $1$ .

$\frac{9 \sqrt{82}}{{\sqrt{82}}^{1} \sqrt{82}}$

Step 4.3

Raise $\sqrt{82}$ to the power of $1$ .

$\frac{9 \sqrt{82}}{{\sqrt{82}}^{1} {\sqrt{82}}^{1}}$

Step 4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{9 \sqrt{82}}{{\sqrt{82}}^{1 + 1}}$

Step 4.5

Add $1$ and $1$ .

$\frac{9 \sqrt{82}}{{\sqrt{82}}^{2}}$

Step 4.6

Rewrite ${\sqrt{82}}^{2}$ as $82$ .

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Step 4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{82}$ as $82^{\frac{1}{2}}$ .

$\frac{9 \sqrt{82}}{{(82^{\frac{1}{2}})}^{2}}$

Step 4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{9 \sqrt{82}}{82^{\frac{1}{2} \cdot 2}}$

Step 4.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{9 \sqrt{82}}{82^{\frac{2}{2}}}$

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

$\frac{9 \sqrt{82}}{82^{\frac{2}{2}}}$

Step 4.6.4.2

Rewrite the expression.

$\frac{9 \sqrt{82}}{82^{1}}$

Step 4.6.5

Evaluate the exponent.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.99388373 \dots$