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Express (e^(pi)/(2)i)e^((pi)/(3)i) in the form a + bi.

October 17, 2026Mathematics

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Answer

\frac{\sqrt{3}}{2} + \frac{1}{2}i

5 steps

Simplify the exponential expression using the rule $\frac{e^x}{e^y} = e^{x-y}$ .
$\frac{e^{\frac{\pi}{2}i}}{e^{\frac{\pi}{3}i}} = e^{\left(\frac{\pi}{2} - \frac{\pi}{3}\right)i}$
Calculate the difference in the exponents.
$\frac{\pi}{2} - \frac{\pi}{3} = \frac{3\pi}{6} - \frac{2\pi}{6} = \frac{\pi}{6}$
So the expression becomes:
$e^{\frac{\pi}{6}i}$
Apply Euler's formula, which states that $e^{i\theta} = \cos(\theta) + i\sin(\theta)$ .
Here, $\theta = \frac{\pi}{6}$ .
$e^{\frac{\pi}{6}i} = \cos\left(\frac{\pi}{6}\right) + i\sin\left(\frac{\pi}{6}\right)$
Evaluate the trigonometric values for $\theta = \frac{\pi}{6}$ .
$\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$ $\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$
Substitute these values back into the expression to get the form $a+bi$ .
$e^{\frac{\pi}{6}i} = \frac{\sqrt{3}}{2} + i\frac{1}{2}$
The expression in the form $a+bi$ is:
$\frac{\sqrt{3}}{2} + \frac{1}{2}i$

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