Before we delve into the concept of exponents, consider the following question as a way to motivate the topic:
Write the number 2187 as a product of its prime factors
Okay, that's not TOO difficult. First, you need to recall the definition of prime factor: a number which is both a factor and which is prime. So, 2187 has a prime factor of 3, and so 2187 written as a product of its prime factors is:
3×3×3×3×3×3×3=2187
But, what if you were given the following question, instead:
Write the number 268435456 as a product of its prime factors
First of all, you would NEVER be asked such a question on the GRE exam ... but bear with us! So, assume that you know that 268435456 is divisible by 2, and so 2 is the smallest prime factor of 268435456. Then, 268435456 written as a product of its prime factors is:
2×2×2×2×2×2×2× ... 2×2×2×2, a total of 28 times = 268435456
And so that's where the concept of exponents comes into play. Exponential notation is an easier way to write a number as a product of many factors. The base of an expression is the number that is being manipulated, and the exponent is the value to which the base number is raised to.
For example, in xa, x is the base and a is the exponent. So, using the exponent notation,
268435456 as a product of its prime factors = 228 = 268435456
There are a few rules that you need to know of when dealing with exponents:
Rule | Example |
Any number raised to the zero power is equal to 1. | 3472^{0} = 1 |
Any number raised to the first power is always equal to itself. | 27^{1} = 27 |
If a number is raised to the second power, we say that the number is squared | 4^{2} is read as "4 squared" |
If a number is raised to the third power, we say that the number is cubed | 3^{3} is read as "3 cubed" |
The types of questions that you might encounter on the GRE exam, and which utilize the concept of exponents, will most often ask you to divide or multiple two exponent quantities. For those types of questions, you'll need to remember the following rules:
Rule | Example |
An exponent raised to some power is the same as the product of the power and the raise: (n^{a})^{b} = n^{ab} | (3^{2})^{3} = 3^{6} = 729 |
If two numbers of the same base but different exponents are multiplied, you add the exponents: n^{a} × n^{b} = n^{a+b} | 3^{2}×3^{3} = 3^{2+3} = 3^{5} |
If dividing two numbers of the same base, subtract the exponent of the denominator from the exponent of the numerator: n^{a} / n^{b} = n^{a-b} | 2^{20} / 2^{18} = 2^{20-18} = 2^{2} = 4 |