proof of (a*b)^2 = a^2*b^2?

Question

how can you prove that (a*b)^2 = a^2*b^2

Answer 1

You can prove (ab)^n = a^n * b^n for all n, using induction. It's been done for n=2 by lots of people here, the next step is proving that if it's true for a given n, it's also true for n+1.

So, given (ab)^n = a^n * b^n for some n, we have
(ab)^(n+1)
= (ab)^n * (ab)^1 (expanding)
= a^n * b^n * ab (by hypothesis)
= a^n * a * b^n * b (rearranging factors)
= a^(n+1) * b^(n+1).
QED.

Answer 2

Simple.

First - squaring means multiplying by itself

(a * b) * (a * b)

As it is all multiplication the brackets aren't needed

a * b * a * b

You can rearrange the terms

a * a * b * b

Again, looking at what squaring means. This is the same as a^2 * b^2

Answer 3

Go to class 5

Answer 4

Take examples as
a=2, b=3 then find
LHS -
a*b= 2*3=6
(6)^2=36 ---------{1}
RHS-
(a)^2=2^2=4
(b)^2=3^2=9
(a)^2*(b)^2= 4*9=36-----------{2}
Therefore {1} ={2}

Answer 5

LHS
=(a*b)^2
=(a*b)*(a*b)
=a*b*a*b
=a*a*b*b
=a^2*b^2
=RHS (proven)

Answer 6

(ab)^2=(ab)(ab)
rearrange the terms
=aabb=a^2 b^2

Answer 7

If greek please e-mail me!!!

(a*b)^2=a^2*b^2

(a*b)^2
=(a*b)*(a*b)
=a*b*a*b
=a^2*b^2