Exponets,how do you combine them?

Post Reply New Topic

Someone posted about having trouble with fractions, and it reminded me of something Ive wondered about exponets. I was probably taugh the answer to this sometime in school but forgot.

Anyway:

What is ( X^N)^N ?

That is: what is x raised to the power of N ... itsself raised to the power of N?


Is it X^N+N? Or is X^NxN ? ( X^2N, or X^N^2 ?).

Do you add the exponets, or do you multiply the exonets in that situation?

Look at it this way:
Temp=X^N
Answer=Temp^N

Evaluate expressions in parenthesis first.

Let a be an arbitrary base and m and n be an arbitrary exponents, then

(a^n)^m = a^(n*m)
(a^n)*(a^m) = a^(n+m)
(a^n)*(a^-m) = a^(n-m)
a^(n/m) = msqrt(a^n)
a^(-n) = 1/a^n
a^m = m*ln(a)
a^0 = 1

A good way to remember this is to look at an example of why it works.

x^2 = xx
x^3 = xxx

(x^3)^2 = (xxx)(xxx) = x^6 = x^(2*3) = x^(3+3)

(x^2)^3 = (xx)(xx)(xx) = x^6 = x^(3*2) = x^(2+2+2)

(x^2)(x^3) = (xx)(xxx) = x^5 = x^(2+3)

You start with a bunch of x's multiplied together, and you count them. If you multiply by another bunch of x's (maybe a different length), you make a longer string of x's, and counting them is the same as adding the two original numbers.

Taking something to a power means repeatedly multiplying the same thing. So you put a bunch of identical strings of x's together, and since multiplication is repeated addition and the exponents add (repeatedly in this case), it is the same as multiplying the exponents together.