What is 5e^{pi/4 i} ?

The solution to 5e^{pi/4 i} is (5sqrt(2))/2+i(5sqrt(2))/2

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simplify 5e^{pi/4 i}

simplify

5 e^{\frac{π}{4} i}

\frac{5 2}{2} + i \frac{5 2}{2}

Solution steps

5 e^{\frac{π}{4} i}

e^{\frac{π}{4} i} = \frac{2 + 2 i}{2}

= 5 \cdot \frac{2 + 2 i}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{( 2 + 2 i ) \cdot 5}{2}

Factor

2 + 2 i : 2 (1 + i)

= \frac{2 ( 1 + i ) \cdot 5}{2}

Cancel

\frac{2 ( 1 + i ) \cdot 5}{2} : \frac{( 1 + i ) \cdot 5}{2}

= \frac{( 1 + i ) \cdot 5}{2}

Rationalize

\frac{5 ( 1 + i )}{2} : \frac{5 2 ( 1 + i )}{2}

= \frac{5 2 ( 1 + i )}{2}

Rewrite

\frac{5 2 ( 1 + i )}{2}

in standard complex form:

\frac{5 2}{2} + \frac{5 2}{2} i

= \frac{5 2}{2} + \frac{5 2}{2} i

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