I'm doing an Advanced Functions course right now, and I'm wondering about something. Look at this here evaluation/simplification that I did:
What I'm wondering is about the 2^4 – 2^3. I know this is is 2^3 because 2^3 is the value multiplied by 2 to get 2^4, which means I'm subtracting the same value I'm adding when going from 2^3 to 2^4 from 2^3 and thereby going back to 2^3.
But this is just a logical step I did it my head, and it's not something that works as a rule for all addition/subtraction. So I'm wondering if there is another way for me to solve this? I'm not comfortable with simplifying something on the premise of "I just know that's what it means", because that might mean I don't get the full score. So is there any exponent rules/laws that apply to adding or subtracting exponents with the same base which I can use here, in order to make sure I don't get less point for taking shortcuts instead of using the exponent laws?
Should I perhaps make it into 2^7 – 2^6 and evaluate it as 128 – 64 = 64? Or is there another way to end up with 2^6 using an exponent law? I just want to make sure I don't get any less points for doing steps in my head or simplifying something outside of the methods of exponent laws.
Best Answer
All you can really say is this. Suppose that $m>n$ and we are looking at
$$a^m+a^n.$$ By definition $m=n+k$ so both have a common factor of $a^n$:
$$a^m+a^n=a^{n+k}+a^n=a^{n}(a^k+1).$$
In your example,
$$2^4-2^3=2^3(2-1)=2^3.$$
However to be honest... $2^4=16$ and $2^{3}=8$ --- it is as simple as that.
Or a bit more general but closer to your example: $$a^{m+1}-a^{m}=a^m(a-1).$$