Base
In the expression b^n, the variable b represents the base. This is the number that we multiply by itself n times.
Exponent
In the expression b^n, the variable n represents the exponent. The exponent indicates how many times to multiple the base, b, by itself. For example, 4^3 = 4 * 4 * 4, or 4 multiplied by itself three times.
Exponential Equations
Equations that include an exponent. x^2 = 16 is an exponential equation. When solving equations with even exponents, we must consider both positive and negative possibilities for the solutions. For x^2 = 16, the two possible solutions are 4 and -4.
Base of Zero
An exponential expression with base 0 yields 0, regardless of the exponent. 0^15 = 0. (Note: the case of 0^0 is not tested on the GMAT.)
Base of One
An exponential expression with base 1 yield 1, regardless of the exponent. 1^15 = 1.
Base of Negative One
An exponential expression with base -1 yields 1 when the exponent is even and -1 when the exponent is odd. (-1)^15 = -1. (-1)^16 = 1.
Fractional Base
When the base is a fraction between zero and one, the value decreases as the exponent increases. (1/2)^3 = 1/2 * 1/2 * 1/2 = 1/8, which is smaller than the starting fraction, 1/2.
Compound Base
When the base represents a product (multiplication) or quotient (division), we can choose to multiply or divide the base first and then raise the base to the exponent, or we can distribute the exponent to each number in the base. (3 * 4)^2 = 12^2 = 144. (3*4)^2 = 3^2 * 4^2 = 9 * 16 = 144.
Exponent of Zero
Any nonzero base raised to the 0 power yields 1. 15^0 = 1.
Exponent of One
Any based raised to the exponent of 1 yields the original base. 15^1 = 15.
Manipulating terms with the same base
| Rule |
Result |
| 4^4 * 4^3 -> Add the exponents |
4^7 |
| 4^4 / 4^3 -> Subtract the exponents |
4^1 |
| (4^4)^3 -> Multiply the exponents |
4^12 |
Negative Exponents
Put the term containing the exponent in the denominator of a fraction and make the exponent positive. 4^-2 = (1/4)^2. (1/3)^-2 = [1/(1/3)]^2 = 3^2 = 9.
Fractional Exponents
If the exponent is a fraction, the numerator reflects what power to raise the base to, and the denominator reflects which root to take. 4^(2/3) = CUBEROOT(4^2).
