Are those exponential expressions equivalent?

Question

5^{25m^2}*5^{m^2}

25^{5m^2+5m}

Are they equivalent?

Answer 1

No.

The first exponent is 25m².

The second exponent is m².

Because the two initial bases are both 5, we may add the exponents getting

5^{25m^2 + 5m^2} = 5^{30m^2}

Alternately, and more simply we can substitute in m =1 to demonstrate they are not the same.

5²⁵⁽¹⁾ * 5¹ = 5²⁶ which is not the same as 25¹⁰ = 25^{5(1) + 5}

See ** below.

Watch for a subtle and very obsure error. When dealing with exponent of exponents, it's important to identify where the higher level exponent is applying.

5 ^{m^2} I'm interpreting as 5 ^{(m^2)} where the square only applies to m. This is very different from ( 5 ^{m ) ^2} = (5 ^m) * (5 ^m) where the square applies to the entire 5^m

Try with m = 2 to prove they are different.

** Special note: Examples, like m=1, can prove a statement false. However, examples can never prove a statement with variables or a theorem true.