Nick H. answered • 03/29/19

Math and Computer Tutor, Patient in Central Arlington

No.

The first exponent is 25m^{2}.

The second exponent is m^{2}.

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.

5^{25(1)} * 5^{1} = 5^{26} which is not the same as 25^{10 }= 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.