Multiplication and Division of Exponents Notes By: Tamiya, Chris, Shelby, Qua von

Multiplication Rule X n. X m =X n+m X n. X m =X n+m In order to use this rule the base numbers being multiplied must be the same Example: X 4. X 4 Written in multiplication form X. X. X. X. X. X. X Using form X 3+4 =X 7

Example *2*2 2*2*2*2*2 2*2*2 2*2*2*2*

Example 2 x 3. x 5. x 4 x 3. x 5. x 4 x x x 12 x 12

Example 3 x. x 4. y 4. z x. x 4. y 4. z Remember x=x 1 Remember x=x 1 x 1+4 y 4 z x5y4zx5y4zx5y4zx5y4z

Example 4 (-2x 3 y 5 )(3xy 3 )(x 2 y) Multiply coefficients and add exponents of like bases Multiply coefficients and add exponents of like bases -6x (3+1+2) y (5+3+1) -6x (3+1+2) y (5+3+1) -6x 6 y 9 -6x 6 y 9

Example 5 4x 5 (-2x 2 y+5xy 2 ) 4x 5 (-2x 2 y+5xy 2 ) In order to simplify you must distribute. Since you are multiplying when you distribute you must use the multiplication rule for exponents. 4x 5 * -2x 2 y + 4x 5 * 5xy 2 -8x 5+2 y + 20x 5+1 y 2 -8x 7 +20x 6 y 2

Dividing Exponents + You can divide exponential expressions, leaving the answers as exponential expressions, as long as the bases are the same. To divide exponents (or powers) with the same base, subtract the exponents. Division is the opposite of multiplication, so it makes sense that because you add exponents when multiplying numbers with the same base, you subtract the exponents when dividing numbers with the same base.

Dividing Exponents Example 2 10 /2 4 = =2 6 Any number to the power of zero equals 1, as long as the base number is not 0

Try It Yourself! = = /2 3 = /4 9 = /5 2 =

The Answers = = /2 3 = /4 9 =4 0 = = /5 2 =5 2

Almost Finish We need to address Powers Property

Power of a Power Property To find a power of a power, multiply the exponents. (5 2 )4 = = 5 8

Power of a Power Property You Try!! (y2)4 [(-3 3 )] 2 [(a+1) 2 ] 5

Power of a Power Property You Try!! (y2)4 = y24 = y8 [(-3 3 )] 2 = (-3) 32 = (-3) 6 [(a+1) 2 ] 5 = (a+1) 25 = (a+1) 10

Power of a Product Property To find a power of a product, find the power of each factor and multiply. (2 3) 6 = = 64 x 729 = 46,656 -(2w) 2 = -(2 w) 2 = -(2 2 w 2 ) = -4w 2

Power of a Product Property You Try (6 5) 2 =

Power of a Product Property You Try (6 5) 2 = = = = 900 = 900

Power of a Product Property You Try (4yz)3 =

Power of a Product Property You Try (4yz)3 = ( 4 y z) 3 = 4 3 y 3 z 3 = 4 3 y 3 z 3 = 64y 3 z 3 = 64y 3 z 3

Using All Properties (expect division) You Try (4x 2 y) 3 x 5

Using All Properties (expect division) You Try (4x 2 y) 3 x 5 = 4 3 (x 2 ) 3 y 3 x 5 = 4 3 (x 2 ) 3 y 3 x 5 = 64 x 6 y 3 x 5 = 64 x 6 y 3 x 5 = 64x 11 y 3 = 64x 11 y 3

THE END