Exponents They may look like numbers, but they don't act like numbers

Peer Review : Reviewed by : Jimbo Jones This presentation was good, but incomplete. It would benefit from more examples of how exponent operations are different from non-exponent operations. The writing is so good, I find it hard to believe it was not taken from the internet. Author responds: I dumbed down the words a bit, in order to sound less fancy.

Exponents (also called "powers") are defined in four different ways: Positive whole number exponents (i.e. 4 3 ) mean “repeated multiplication”. ( 4 3 = 4 x 4 x 4 = 64 ) Negative exponents (i.e. 5 –3 ) mean “repeated division”, or “bring it into a fraction (in the opposite place, either bottom or top). ( 5 –3 = 1 ÷ 5 ÷ 5 ÷ 5 = 1 / 125 ) Fraction exponents (i.e. 9 ½ ) mean “roots”. ( 9 ½ = square root of 9 = 3 ) A zero exponent is defined to equal the number 1, for ANY non-zero base. Basic Facts

Exponents do not follow the normal rules for operations, but they do follow the rules for the operations that are “one level down”. Operations with Exponents This is a major reason that some people have trouble performing operations with exponents

When two same-letter terms with exponents are MULTIPLIED, the exponents get ADDED. For example, x 4 x 5 = x 9 ( not x 20, even though 4 5 = 20 ) Even though it is a multiplication problem, the exponents get added (addition is “one level down” from multiplication).

When a term with exponents gets raised to a power (another exponent), the two exponents get MULTIPLIED. For example, (x 4 ) 3 = x 12, ( not x 64, even though 4 3 = 64 ) Even though it is an exponent problem, the exponents get multiplied (multiplication is “one level down” from exponents).

When two same-letter terms with exponents are DIVIDED, the exponents get SUBTRACTED. For example, x 6 ÷ x 2 = x 4 ( not x 3, even though 6 ÷ 2 = 3 ) Even though it is a division problem, the exponents get subtracted (subtraction is “one level down” from division).

When a term with exponents is inside a root (such as a square root), the exponents and the index get DIVIDED. For example, √ (x 16 ) = x 8, (not x 4, even though √ 16 = 4) Even though it is a square root problem, the exponents get divided (division is “one level down” from exponents).

This becomes a challenge when you have operations involving coefficients, variables, and exponents. Coefficients are numbers. They are multipliers of the variables. Coefficients behave like numbers. Exponents behave "one level down". For example, in (2a 3 b 5 ) 4, the 2 is a coefficient, so it gets raised to the fourth power. (2 4 = 16). The 3 and the 5 are exponents, so they get multiplied by the outside exponent of 4. (2a 3 b 5 ) 4 = 16a 12 b 20, not 8a 12 b 20, nor 16a 81 b 625