Trigonometry Examples

$\sqrt{\frac{21}{2} \cdot ((\frac{21}{2} - 5) (\frac{21}{2} - 7) (\frac{21}{2} - 9))}$

Step 1

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\sqrt{\frac{21}{2} \cdot (\frac{21}{2} - 5 \cdot \frac{2}{2}) (\frac{21}{2} - 7) (\frac{21}{2} - 9)}$

Step 2

Combine $- 5$ and $\frac{2}{2}$ .

$\sqrt{\frac{21}{2} \cdot (\frac{21}{2} + \frac{- 5 \cdot 2}{2}) (\frac{21}{2} - 7) (\frac{21}{2} - 9)}$

Step 3

Combine the numerators over the common denominator.

$\sqrt{\frac{21}{2} \cdot \frac{21 - 5 \cdot 2}{2} (\frac{21}{2} - 7) (\frac{21}{2} - 9)}$

Step 4

Simplify the numerator.

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Step 4.1

Multiply $- 5$ by $2$ .

$\sqrt{\frac{21}{2} \cdot \frac{21 - 10}{2} (\frac{21}{2} - 7) (\frac{21}{2} - 9)}$

Step 4.2

Subtract $10$ from $21$ .

$\sqrt{\frac{21}{2} \cdot \frac{11}{2} (\frac{21}{2} - 7) (\frac{21}{2} - 9)}$

Step 5

Combine exponents.

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Step 5.1

Multiply $\frac{21}{2}$ by $\frac{11}{2}$ .

$\sqrt{\frac{21 \cdot 11}{2 \cdot 2} (\frac{21}{2} - 7) (\frac{21}{2} - 9)}$

Step 5.2

Multiply $21$ by $11$ .

$\sqrt{\frac{231}{2 \cdot 2} (\frac{21}{2} - 7) (\frac{21}{2} - 9)}$

Step 5.3

Multiply $2$ by $2$ .

$\sqrt{\frac{231}{4} (\frac{21}{2} - 7) (\frac{21}{2} - 9)}$

Step 6

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\sqrt{\frac{231}{4} (\frac{21}{2} - 7 \cdot \frac{2}{2}) (\frac{21}{2} - 9)}$

Step 7

Combine $- 7$ and $\frac{2}{2}$ .

$\sqrt{\frac{231}{4} (\frac{21}{2} + \frac{- 7 \cdot 2}{2}) (\frac{21}{2} - 9)}$

Step 8

Combine the numerators over the common denominator.

$\sqrt{\frac{231}{4} \cdot \frac{21 - 7 \cdot 2}{2} (\frac{21}{2} - 9)}$

Step 9

Simplify the numerator.

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Step 9.1

Multiply $- 7$ by $2$ .

$\sqrt{\frac{231}{4} \cdot \frac{21 - 14}{2} (\frac{21}{2} - 9)}$

Step 9.2

Subtract $14$ from $21$ .

$\sqrt{\frac{231}{4} \cdot \frac{7}{2} (\frac{21}{2} - 9)}$

Step 10

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\sqrt{\frac{231}{4} \cdot \frac{7}{2} (\frac{21}{2} - 9 \cdot \frac{2}{2})}$

Step 11

Combine $- 9$ and $\frac{2}{2}$ .

$\sqrt{\frac{231}{4} \cdot \frac{7}{2} (\frac{21}{2} + \frac{- 9 \cdot 2}{2})}$

Step 12

Combine the numerators over the common denominator.

$\sqrt{\frac{231}{4} \cdot \frac{7}{2} \frac{21 - 9 \cdot 2}{2}}$

Step 13

Simplify the numerator.

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Step 13.1

Multiply $- 9$ by $2$ .

$\sqrt{\frac{231}{4} \cdot \frac{7}{2} \frac{21 - 18}{2}}$

Step 13.2

Subtract $18$ from $21$ .

$\sqrt{\frac{231}{4} \cdot \frac{7}{2} \frac{3}{2}}$

Step 14

Combine exponents.

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Step 14.1

Multiply $\frac{231}{4}$ by $\frac{7}{2}$ .

$\sqrt{\frac{231 \cdot 7}{4 \cdot 2} \cdot \frac{3}{2}}$

Step 14.2

Multiply $231$ by $7$ .

$\sqrt{\frac{1617}{4 \cdot 2} \cdot \frac{3}{2}}$

Step 14.3

Multiply $4$ by $2$ .

$\sqrt{\frac{1617}{8} \cdot \frac{3}{2}}$

Step 14.4

Multiply $\frac{1617}{8}$ by $\frac{3}{2}$ .

$\sqrt{\frac{1617 \cdot 3}{8 \cdot 2}}$

Step 14.5

Multiply $1617$ by $3$ .

$\sqrt{\frac{4851}{8 \cdot 2}}$

Step 14.6

Multiply $8$ by $2$ .

$\sqrt{\frac{4851}{16}}$

Step 15

Rewrite $\sqrt{\frac{4851}{16}}$ as $\frac{\sqrt{4851}}{\sqrt{16}}$ .

$\frac{\sqrt{4851}}{\sqrt{16}}$

Step 16

Simplify the numerator.

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Step 16.1

Rewrite $4851$ as $21^{2} \cdot 11$ .

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Step 16.1.1

Factor $441$ out of $4851$ .

$\frac{\sqrt{441 (11)}}{\sqrt{16}}$

Step 16.1.2

Rewrite $441$ as $21^{2}$ .

$\frac{\sqrt{21^{2} \cdot 11}}{\sqrt{16}}$

Step 16.2

Pull terms out from under the radical.

$\frac{21 \sqrt{11}}{\sqrt{16}}$

Step 17

Simplify the denominator.

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Step 17.1

Rewrite $16$ as $4^{2}$ .

$\frac{21 \sqrt{11}}{\sqrt{4^{2}}}$

Step 17.2

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

$\frac{21 \sqrt{11}}{4}$

Step 18

The result can be shown in multiple forms.

Exact Form:

$\frac{21 \sqrt{11}}{4}$

Decimal Form:

$17.41228014 \dots$