So now you know how to combine exponential expressions when they share a common base. But what can you do when presented with an expression such as 53 × 252? At first, it may seem that no further simplification is possible.
The trick here is to realize that 25 is actually 52. Because they are equivalent values, you can replace 25 with 52 and see what you get.
You can write 53 × (52)2 as 53 × 54. This expression can now be combined and you end up with 57.
When dealing with exponential expressions, you need to be on the lookout for perfect squares and perfect cubes that can be rewritten. In the last example, 25 is a perfect square and can be rewritten as 52. In general, it is good to know all the perfect squares up to 152, the perfect cubes up to 63, and the powers of 2 and 3. Here’s a brief list of some of the numbers likely to appear on the GRE:
The powers of 2: 2, 4, 8, 16, 32, 64, 128
The powers of 3: 3, 9, 27, 81
| 42 = 16 52 = 25 62 = 36 72 = 49 82 = 64 92 = 81 |
102 = 100 112 = 121 122 = 144 132 = 169 142 = 196 152 = 225 |
23 = 8 33 = 27 43 = 64 53 = 125 |
Let’s try another example. How would you combine the expression 23 × 84? Try it out for yourself.
Again, the key is to recognize that 8 is 23. The expression can be rewritten as 23 × (23)4, which becomes 23 × 212, which equals 215.
Alternatively, you could replace 23 with 81. The expression can be rewritten as 81 × 84, which equals 85.
Combine the following expressions.
24 × 163
75 × 498
93 × 813