Simplifying Exponential Expressions

Now that you have the basics down for working with bases and exponents, what about working with multiple exponential expressions at the same time? If two (or more) exponential terms in an expression have a base in common or an exponent in common, you can often simplify the expression. (In this section, “simplify,” means “reduce to one term.”)

When Can You Simplify Exponential Expressions?

  1. You can only simplify exponential expressions that are linked by multiplication or division. You cannot simplify expressions linked by addition or subtraction (although, in some cases, you can factor them and otherwise manipulate them).
  2. You can simplify exponential expressions linked by multiplication or division if they have either a base or an exponent in common.

How Can You Simplify Them?

Use the exponent rules described earlier. If you forget these rules, you can derive them on the test by writing out the example exponential expressions. For example:

These expressions
CANNOT be simplified:		 These expressions CAN
be simplified:		 Here’s how:
74 + 76 (74)(76) (74)(76) = 74 + 6 = 710
34 + 124 (34)(124) (34)(124) = (3 × 12)4 = 364
65 − 63
127 − 37

You can simplify all the expressions in the middle column to a single term, because the terms are multiplied or divided. The expressions in the left column cannot be simplified, because the terms are added or subtracted. However, they can be factored whenever the base is the same. For example, 74 + 76 can be factored because the two terms in the expression have a factor in common. What factor exactly do they have in common? Both terms contain 74. If you factor 74 out of each term, you are left with 74(72 + 1) = 74(50).

The terms can also be factored whenever the exponent is the same and the terms contain something in common in the base. For example, 34 + 124 can be factored because 124 = (2 × 2 × 3)4. Thus, both bases contain 34, and the factored expression is 34(1 + 44) = 34(257).

Likewise, 65 − 63 can be factored as 63(62 − 1). 63(35) and 127 − 37 can be factored as 37(47 − 1).

On the GRE, it generally pays to factor exponential terms that have something in common in the bases. For example:

If x = 420 + 421 + 422, what is the largest prime factor of x?

All three terms contain 420, so you can factor the expression: x = 420(40 + 41 + 42). Therefore, x = 420(1 + 4 + 16) = 420(21) = 420(3 × 7). The largest prime factor of x is 7.