Question

If a = b^x, b = c^y and c = a^z prove that xyz = 1.

Answer 1

Given:
a = b^x

b = c^y

c = a^z

To Prove: xyz = 1

Proof (Stepwise):

1. From the given, substitute (b) in terms of (c) and (y) in the first equation:

a = bx

a = (cy)x

a = c(xy)

2. From the third equation, we have

c = az

3. Substitute (c = az) into the expression for (a):

a = c(xy)

a = az(xy)

4. Divide both sides by (a) assuming (a ≠ 0):

1 = z(xy)

5. Since multiplication is associative and commutative,

1 = xyz

We have proved that xyz = 1.