Exponent Rules

somail wrote:How is the following simplification done:

2^100-2^96

We can only add or subtract in these situations if the base is the same AND the exponent is the same.

For example, we can solve:

5(x^5) - 2(x^5) = 3(x^5)

However, there's no simple solution for:

5(x^5) - 2(x^3)

So, if we're faced with a situation in which the bases are the same but the exponents aren't, we need to find a common exponent. We do so through factoring.

Let's look at the original question:

2^100 - 2^96

To get the same exponent, we need to reduce 2^100.

We can rewrite 2^100 as (2^4)(2^96) = 16(2^96).
We can rewrite 2^96 as 1(2^96).

So, we can rewrite 2^100 - 2^96 as:

16(2^96) - 1(2^96) = 15(2^96)

Wow. Thank you so much. Thats been bugging me all morning.