A Number Raised to the m Power

[Bashyboy]

Hello Everyone,

I was wondering, does anyone know of a proof that showed if a number is raised to the mth power, where m is a positive even number, the number is always real?

 

[Borek]

(1+2i)⁴ = -7-24i

 

[Bashyboy]

Drats! I was hoping it was true! How about if m were only 2? Would the statement then be true?

 

[Bashyboy]

It isn't: (a + bi)(a + bi) = a^2 + 2abi + b^2i^2 = a^2 + 2abi - b^2

 

[DrewD]

Drats! I was hoping it was true! How about if m were only 2? Would the statement then be true?

$(1+i)^2=1+2i-1=2i$

Your claim will only be true for $m=2$ if the number is only real or imaginary. This is clear by expanding a complex number as a binomial.

$(a+bi)^2=a^2-b^2+2abi$

In order for this to be real, either $a$ or $b$ must be zero. You should check the case when $m=4$ by squaring $(a^2-b^2+abi)$ to see what you get.


Edit: I see that you figured it out as I was posting