Expert: Paul Klarreich - 2/7/2008

Question
Hi
I really need help solving this proof.
Prove by induction on n, that a^-n= (a^n)^-1 for any positive integer n.

Answer
Questioner:   Jen
Category:  Calculus
Private:  No
 
Subject:  Prove by induction
Question:  Hi
I really need help solving this proof.
Prove by induction on n, that a^-n= (a^n)^-1 for any positive integer n.
........................................
Hi, Jen,

It is hard to know, from what you wrote, just what facts you are allowed to use.  So I will

assume you can use only the IMPORTANT

FACT1 that   x^-1 = 1/x
FACT2 that   x^a x^b = x^(a+b)

Proof by induction:

Base case:  n = 1:

a^-1 = (a)^-1 = (a^1)^-1, done.

Assumption: n = k is true:

a^-k = (a^k)^-1   << is assumed

To Prove:  n = k+1 is true.

a^-(k+1) = (a^(k+1))^-1    << to prove.

a^-(k+1) = a^(-k-1)  << meaning of  -(...)
= a^-k a^-1    << rule for sum of exponents.
= (a^k)^-1 a^-1  << ASSUMPTION USED HERE.
= 1/(a^k) 1/a    << use of FACT1.

= 1/(a^k a)          << mutlplication of fractions.
= 1/(a^k a^1)    << def of  a^1
= 1/(a^k+1)      << use FACT2
= (a^(k+1))^-1     << use FACT1.

-- done.