Better understanding of advanced exponential technique

[tiffany.a.scott]

I am using the 5lb Book of GRE Practice Problems from 2013.

I would just like to understand the exponential technique used in Chapter 7 - Arithmetic - Question 7.

When it says, "Alternatively, you could factor out 5^2 (this is an important technique for larger numbers and exponents where pure arithmetic would be impractical):

5^3-5^2=
5^2(5^1-1)=
5^2(4)=
100
"

I would like to know how to do this, because I'm just not understanding from this example. I have come to understand the method with using the same exponents like 2^2+2^2+2^2-3^2+3^2 = 2^2(1+1+1) or 2^2(3)-3^2(1+1) or 2^2(3)-3^2(2) and simplify if needed, but I am not sure how to go about the above example or others like it by using the advanced method above.

[narang.priyansh]

Consider looking at this way:

"x^y is nothing but x multiplied by itself y times" so 5^3 is 5 multiplied by itself 3 times or 5*5*5, similarly 5^2 is 5 multiplied by itself 2 times or 5*5. Now, if you were to find a difference between the two products, the numbers may be bigger than in this question so going back to the basics of factors and multiples - if 5 is multiplied by itself thrice then it is definitely multiplied by itself twice hence from the first and second term you can see 5*5 becomes a factor. If you take that factor out, you'll be left with a 5 and a 1 which need to be subtracted. This leads to your answer being
=5^2(5-1)
=25*4
=100

As a general rule for exponents: x^n where x is the base and n is the exponent power, x^0, x^1, x^2....x^n are all its factors. You can simplify your expression using this property.

[tommywallach]

That's a great explanation. I'll just jump in as well. Remember you can always factor out any FACTOR that's in every term in the expression. For example.

4 + 8 --> 2 (2 + 4)
4 + 8 --> 4 (1 + 2)

In the first example, I factored out a 2, because there is a factor of 2 in both 4 and 8.
In the second example, I factored out a 4, because there is a factor of 4 in both 4 and 8.
(In general, you want to factor out the LARGEST shared factor, as in my second example; rarely is it more helpful to factor out a small shared factor, though it is SOMETIMES better.)

Now, let's look at your example.

5^3-5^2=

For a second, let's look at it as just numbers:

125 - 25=

What is the biggest shared factor of 125 and 25? It's 25 (i.e. 5^2). So we can factor that out:

5^2 (5 - 1)

Make sense?

-t