1. Complex number

What is 3^(3+ i(8pi/ln3))

Work done so far

3^3 * 3^( i8pi/ln(3) ) = 27 * (3^ i8pi - 3^ln(3) ) = 27 *( 3^25.1i -3.34)

3^(3+ i25.1) - 90.26

I am not sure how to convert i to a number.........

2. Re: Complex number

Originally Posted by mathsforumhelp
What is 3^(3+ i(8pi/ln3))
I don't know how advanced you are.
In complex numbers $z^w=\exp(w\cdot \log(z))$.
Depending upon your level, you may want to use only the principal log.
If so, we get $\exp \left( {\ln (3)\left( {3 + \frac{{8\pi i}}{{\ln (3)}}} \right)} \right) = \exp \left( {\ln (27) + 8\pi i} \right) = 27$

3. Re: Complex number

Hello, mathsforumhelp!

$\text{Evaluate: }\:X \;=\;3^{3+ \frac{8\pi}{\ln(3)}i}$

We have: . $X \;=\;3^3\cdot3^{\frac{8\pi}{\ln(3)}i} \;=\;27\cdot3^{\frac{8\pi}{\ln(3)}i}$ .[1]

Let $y \:=\:3^{\frac{8\pi}{\ln(3)}i}$

Take logs: . $\ln(y) \:=\:\ln\left(3^{\frac{8\pi}{\ln(3)}i}\right) \:=\:\frac{8\pi}{\ln(3)}i\cdot\ln(3) \quad\Rightarrow\quad \ln(y) \:=\:\8\pi i$

Hence: . $y \:=\:e^{8\pi i} \:=\:(e^{i\pi})^8 \:=\:(\text{-}1)^8 \:=\:1$

Substitute into [1]: . $X \:=\:27\cdot1 \:=\:27$

4. Re: Complex number

Thanks guys.....