# Math Help - Power Proof

1. ## Power Proof

Hey all, I'd really appreciate some help with the following proof about powers:

(b^m)^k = b^(mk)

Power is defined in the following way. Let b be a fixed integer. We define b^k for all integers k >= 0 by:

b^0 := 1

Assuming b^n defined, let b^(n+1) := b^n * b

Thanks a lot!

2. Originally Posted by jstarks44444
Hey all, I'd really appreciate some help with the following proof about powers:

(b^m)^k = b^(mk)

Power is defined in the following way. Let b be a fixed integer. We define b^k for all integers k >= 0 by:

b^0 := 1

Assuming b^n defined, let b^(n+1) := b^n * b

Thanks a lot!
Well pf. by induction.

Base case $k=0$

$(b^m)^0=1$ by definition.

Assume the cases $k$ is true

$(b^m)^k=b^{mk}$

Now show $k \implies k+1$

$(b^{m})^{k+1}=(b^m)^k\cdot (b^m)^1$ by your assumption

Now by the induction hypothesis we get
$b^{mk}\codt b^m=b^{mk+m}=b^{m(k+1)}$