Finite Math Examples

$2^{26 (\cos (\frac{26 π}{3}) + i \sin (\frac{26 π}{3}))}$

Step 1

Simplify each term.

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Step 1.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$2^{26 (\cos (\frac{2 π}{3}) + i \sin (\frac{26 π}{3}))}$

Step 1.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$2^{26 (- \cos (\frac{π}{3}) + i \sin (\frac{26 π}{3}))}$

Step 1.3

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$2^{26 (- \frac{1}{2} + i \sin (\frac{26 π}{3}))}$

Step 1.4

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$2^{26 (- \frac{1}{2} + i \sin (\frac{2 π}{3}))}$

Step 1.5

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$2^{26 (- \frac{1}{2} + i \sin (\frac{π}{3}))}$

Step 1.6

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$2^{26 (- \frac{1}{2} + i \frac{\sqrt{3}}{2})}$

Step 1.7

Combine $i$ and $\frac{\sqrt{3}}{2}$ .

$2^{26 (- \frac{1}{2} + \frac{i \sqrt{3}}{2})}$

Step 2

Simplify terms.

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Step 2.1

Apply the distributive property.

$2^{26 (- \frac{1}{2}) + 26 \frac{i \sqrt{3}}{2}}$

Step 2.2

Cancel the common factor of $2$ .

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Step 2.2.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$2^{26 (\frac{- 1}{2}) + 26 \frac{i \sqrt{3}}{2}}$

Step 2.2.2

Factor $2$ out of $26$ .

$2^{2 (13) \frac{- 1}{2} + 26 \frac{i \sqrt{3}}{2}}$

Step 2.2.3

Cancel the common factor.

$2^{2 \cdot 13 \frac{- 1}{2} + 26 \frac{i \sqrt{3}}{2}}$

Step 2.2.4

Rewrite the expression.

$2^{13 \cdot - 1 + 26 \frac{i \sqrt{3}}{2}}$

Step 2.3

Multiply $13$ by $- 1$ .

$2^{- 13 + 26 \frac{i \sqrt{3}}{2}}$

Step 2.4

Cancel the common factor of $2$ .

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Step 2.4.1

Factor $2$ out of $26$ .

$2^{- 13 + 2 (13) \frac{i \sqrt{3}}{2}}$

Step 2.4.2

Cancel the common factor.

$2^{- 13 + 2 \cdot 13 \frac{i \sqrt{3}}{2}}$

Step 2.4.3

Rewrite the expression.

$2^{- 13 + 13 (i \sqrt{3})}$

$2^{- 13 + 13 i \sqrt{3}}$