Because 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.
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.
Multiplying and Dividing Exponents
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
When you divide terms with exponents and the same bases, just subtract the exponents. Any coefficients are also divided as usual.
a5 ÷ a3 = a2
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
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
The value of a base with an exponent of 0 is always 1.
a0 = 1
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.
A negative exponent works like a positive exponent with a twist. A negative exponent takes the positive exponent and then flips it around so the exponent becomes its reciprocal.
3-3 = 1/33 = 1/27
Roots are also known as radicals. Roots are sort of the opposite of exponents. You square 3 to get 9, the square root of 9 is 3.
Properties of Radicals
Squaring and Square Roots
Squaring a number that is greater than 1, or raising it to a higher power, results in a larger number; squaring a number between 0 and 1 results in a smaller number.
Square root of a number n is a number that, when squared, is equal to n. The square root of a negative number is not a real number.