Exponent rules

The terms "exponent" and "power" are typically used interchangeably to refer to the superscript "n" in bn. For simplicity on this page, the term "exponent" will be used to refer to numbers of the form bn, and power will be used to refer to the superscript "n." "b" is the base.


adding or subtracting exponents


To add or subtract exponents, the exponents must have the same base and the same power.

Example

3(32) + 32 = (3 + 1)(32) = 4(32)

3x5 - 6x5 = (3 - 6)x5 = -3x5

If exponents do not have the same base and the same power, you cannot add or subtract them (not even their coefficients!).


multiplying exponents


To multiply exponents, the exponents must have the same base and/or the same power. To multiply exponents that have the same base, keep the same base and add the powers together. If exponents have different bases, you cannot add their powers.

If the exponents have coefficients attached to their bases, multiply the coefficients together. Coefficients can be multiplied together even if the exponents have different bases.

Example

32 × 33 = 32+3 = 35

4x6 × 5y4 = (4 × 5)x6y4 = 20x6y4

If the exponents have the same power but different bases, you can multiply the bases together first before taking the power of their product.

Example

42 × 62 = (4 × 6)2 = 242 = 576

If exponents have the same power and the same base, you can choose either method to simplify.

Example

52 × 52 = 52+2 = 54 = 625

52 × 52 = (5 * 5)2 = 252 = 625


dividing exponents


To divide exponents, the exponents must have the same base and/or the same power. To divide exponents that have the same base, keep the same base and subtract the power of the denominator from the power of the numerator. If exponents have different bases, you cannot subtract their powers.

If the exponents have coefficients attached to their bases, divide the coefficients. Coefficients can be divided even if the exponents have different bases.

Example

If an exponent has a negative power, you still need to keep the same sign and subtract the power.

Example

If exponents have the same power but different bases, you can divide the bases first before taking the power of the quotient.

Example

If exponents have the same power and same base, they will cancel out to equal 1.

Example

When taking a power to a power, be wary of the parentheses (if there are any). Follow order of operations, as they also apply to exponents.

If the base and first exponent are in parentheses, you can multiply the first and second exponent together. You can also first take the base to the power of the first exponent, then raise that value to the power of the second exponent.

Example

(43)2 = 43×2 = 46 = 4096

(43)2 = 642 = 4096

If the base and first exponent are not in parentheses, take the power of the first exponent to the second, then simplify.

Example

432 = 4(32) = 49 = 262144

3 to the power of 2 takes precedent over 4 to the power of 3 because there are no parentheses surrounding it. Small details can make huge differences in the answer!

With exponents, parentheses can also be used to surround a coefficient and its variable. In this case, distribute the exponent to both the coefficient and variable.

Example

(5x)3 = 53x3 = 125x3

This can also be applied to division:

Example


See also combining like terms, negative exponents, order of operations.