Trigonometry Examples

$\sqrt{\frac{37}{2} \cdot ((\frac{37}{2} - 12) (\frac{37}{2} - 12) (\frac{37}{2} - 13))}$

Step 1

Raise $\frac{37}{2} - 12$ to the power of $1$ .

$\sqrt{\frac{37}{2} \cdot ({(\frac{37}{2} - 12)}^{1} (\frac{37}{2} - 12)) (\frac{37}{2} - 13)}$

Step 2

Raise $\frac{37}{2} - 12$ to the power of $1$ .

$\sqrt{\frac{37}{2} \cdot ({(\frac{37}{2} - 12)}^{1} {(\frac{37}{2} - 12)}^{1}) (\frac{37}{2} - 13)}$

Step 3

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

$\sqrt{\frac{37}{2} \cdot {(\frac{37}{2} - 12)}^{1 + 1} (\frac{37}{2} - 13)}$

Step 4

Add $1$ and $1$ .

$\sqrt{\frac{37}{2} \cdot {(\frac{37}{2} - 12)}^{2} (\frac{37}{2} - 13)}$

Step 5

Combine $\frac{37}{2}$ and ${(\frac{37}{2} - 12)}^{2}$ .

$\sqrt{\frac{37 {(\frac{37}{2} - 12)}^{2}}{2} (\frac{37}{2} - 13)}$

Step 6

To write $- 12$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\sqrt{\frac{37 {(\frac{37}{2} - 12 \cdot \frac{2}{2})}^{2}}{2} (\frac{37}{2} - 13)}$

Step 7

Combine $- 12$ and $\frac{2}{2}$ .

$\sqrt{\frac{37 {(\frac{37}{2} + \frac{- 12 \cdot 2}{2})}^{2}}{2} (\frac{37}{2} - 13)}$

Step 8

Combine the numerators over the common denominator.

$\sqrt{\frac{37 {(\frac{37 - 12 \cdot 2}{2})}^{2}}{2} (\frac{37}{2} - 13)}$

Step 9

Simplify the numerator.

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Step 9.1

Multiply $- 12$ by $2$ .

$\sqrt{\frac{37 {(\frac{37 - 24}{2})}^{2}}{2} (\frac{37}{2} - 13)}$

Step 9.2

Subtract $24$ from $37$ .

$\sqrt{\frac{37 {(\frac{13}{2})}^{2}}{2} (\frac{37}{2} - 13)}$

Step 10

Apply the product rule to $\frac{13}{2}$ .

$\sqrt{\frac{37 \frac{13^{2}}{2^{2}}}{2} (\frac{37}{2} - 13)}$

Step 11

To write $- 13$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\sqrt{\frac{37 \frac{13^{2}}{2^{2}}}{2} (\frac{37}{2} - 13 \cdot \frac{2}{2})}$

Step 12

Combine $- 13$ and $\frac{2}{2}$ .

$\sqrt{\frac{37 \frac{13^{2}}{2^{2}}}{2} (\frac{37}{2} + \frac{- 13 \cdot 2}{2})}$

Step 13

Combine the numerators over the common denominator.

$\sqrt{\frac{37 \frac{13^{2}}{2^{2}}}{2} \cdot \frac{37 - 13 \cdot 2}{2}}$

Step 14

Simplify the numerator.

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Step 14.1

Multiply $- 13$ by $2$ .

$\sqrt{\frac{37 \frac{13^{2}}{2^{2}}}{2} \cdot \frac{37 - 26}{2}}$

Step 14.2

Subtract $26$ from $37$ .

$\sqrt{\frac{37 \frac{13^{2}}{2^{2}}}{2} \cdot \frac{11}{2}}$

Step 15

Evaluate the exponents.

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Step 15.1

Raise $13$ to the power of $2$ .

$\sqrt{\frac{37 \frac{169}{2^{2}}}{2} \cdot \frac{11}{2}}$

Step 15.2

Raise $2$ to the power of $2$ .

$\sqrt{\frac{37 (\frac{169}{4})}{2} \cdot \frac{11}{2}}$

Step 16

Combine $37$ and $\frac{169}{4}$ .

$\sqrt{\frac{\frac{37 \cdot 169}{4}}{2} \cdot \frac{11}{2}}$

Step 17

Multiply $37$ by $169$ .

$\sqrt{\frac{\frac{6253}{4}}{2} \cdot \frac{11}{2}}$

Step 18

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{\frac{6253}{4} \cdot \frac{1}{2} \frac{11}{2}}$

Step 19

Multiply $\frac{6253}{4} \cdot \frac{1}{2}$ .

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Step 19.1

Multiply $\frac{6253}{4}$ by $\frac{1}{2}$ .

$\sqrt{\frac{6253}{4 \cdot 2} \cdot \frac{11}{2}}$

Step 19.2

Multiply $4$ by $2$ .

$\sqrt{\frac{6253}{8} \cdot \frac{11}{2}}$

Step 20

Multiply $\frac{6253}{8}$ by $\frac{11}{2}$ .

$\sqrt{\frac{6253 \cdot 11}{8 \cdot 2}}$

Step 21

Multiply.

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Step 21.1

Multiply $6253$ by $11$ .

$\sqrt{\frac{68783}{8 \cdot 2}}$

Step 21.2

Multiply $8$ by $2$ .

$\sqrt{\frac{68783}{16}}$

Step 22

Rewrite $\sqrt{\frac{68783}{16}}$ as $\frac{\sqrt{68783}}{\sqrt{16}}$ .

$\frac{\sqrt{68783}}{\sqrt{16}}$

Step 23

Simplify the numerator.

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Step 23.1

Rewrite $68783$ as $13^{2} \cdot 407$ .

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Step 23.1.1

Factor $169$ out of $68783$ .

$\frac{\sqrt{169 (407)}}{\sqrt{16}}$

Step 23.1.2

Rewrite $169$ as $13^{2}$ .

$\frac{\sqrt{13^{2} \cdot 407}}{\sqrt{16}}$

Step 23.2

Pull terms out from under the radical.

$\frac{13 \sqrt{407}}{\sqrt{16}}$

Step 24

Simplify the denominator.

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Step 24.1

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

$\frac{13 \sqrt{407}}{\sqrt{4^{2}}}$

Step 24.2

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

$\frac{13 \sqrt{407}}{4}$

Step 25

The result can be shown in multiple forms.

Exact Form:

$\frac{13 \sqrt{407}}{4}$

Decimal Form:

$65.56628325 \dots$