As multiplication can be thought of as repeated addition, you can think of exponents as repeated multiplication. This means that 43 is the same as 4 × 4 × 4 or 64. 43 means multiplying 4 three times. The result is 64. Here, 4 is the base and superscript 3 is the exponent. If you add a variable into this mix, such as 4b3, the base becomes b and the 4 becomes the coefficient.
Positive and Negative Bases / Even and Odd Exponents
- A positive number taken to an even or odd power remains positive.
- A negative number taken to an odd power remains negative.
- A negative number taken to an even power becomes positive.
Any number taken to an even power either remains or becomes positive, and any number taken to an odd power keeps the sign it began with.
Any term with an odd power that results in a negative number will have a negative root, and this is the only possible root for the expression. For example, if a3 = –125, then a = –5. That is, the cube root of –125 is –5.
On the other hand, anytime you have an exponent of 2, you have two potential roots, one positive and one negative, for the expression. For example, if a2 = 64, then a = 8 or –8. So 64 has two possible square roots: either 8 or –8.
Adding and subtracting with exponents
The only catch to adding or subtracting with exponents is that the base and exponent of each term must be the same. So, you can add and subtract like terms. The base and exponent remain the same and that the coefficient is the only number that changes in the equation.
Multiplying and dividing with exponents
1. To multiply terms with exponents and the same bases, add the exponents. If the expression contains coefficients, multiply the coefficients as you normally would.
a3 × a2 = a5
2. When you divide terms with exponents and the same bases, just subtract the exponents. Any coefficients are also divided as usual.
a5 ÷ a3 = a2
3. To multiply exponential terms with different bases, first make sure the exponents are the same. If they are, multiply the bases and maintain the same exponent. Follow the same procedure when you divide terms with different bases but the same exponents.
a5 × b5 = (ab)5
4. When you raise a power to another power, multiply the exponents. If your expression includes a coefficient, take it to the same power.
(a3)5 = a15
5. The value of a base with an exponent of 0 is always 1.
a0 = 1
6. The value of a base with an exponent of 1 is the same value as the base.
a1 = a
If you see a problem with an exponent in fraction form, consider the top number of the fraction (the numerator) as your actual exponent and the bottom number (the denominator) as the root. For example, to solve 2561/4, simply take 256 to the first power (because the numerator of the fraction is 1), which is 256. Then, take the fourth root of 256 (because the denominator of the fraction is 4), which is 4, and that’s your answer.
A negative exponent works like a positive exponent with a twist. A negative exponent takes the positive exponent and then flips the base and exponent around so that together they become the reciprocal.
3-3 = 1/33 = 1/27
When you work with negative exponents, don’t fall for the trick of assuming that the negative exponent somehow turns the original number into a negative number.