7.4 MULTIPLICATION AND EXPONENTS: Base: A number that is multiplied repeatedly. Exponent: A number that shows repeated multiplication. Property: A character or attribute that something has.

GOAL:

An exponent equation has two components: Remember: An exponent equation has two components: 𝑏 𝑥 Exponent Base

For every number a≠0 and m, n, are integers, Raising powers to powers: PROPERTIES: For every number a≠0 and m, n, are integers, (𝑎 𝑚 )𝑛 = 𝑎 𝑚∙𝑛 Ex: 1) (41)3 = 41∙3 = 43 = 64 = 𝟏 𝟑 𝟑 = 𝟏 𝟐𝟕 2) (31)-3 = 31 ∙ -3 = 3-3

YOU TRY IT: Simplify: (124)-2 ((-2)5)-2 (m3 )-1 ∙ m5 (9-3 )2 ∙ 9-4

SOLUTION: No matter what integer it is, anything to the power of zero is 1. (124)-2  𝟏 𝟏𝟐 𝟖  12(4)(-2)  12-8 2) ((-2)5)-2  𝟏 (−𝟐) 𝟏𝟎  (-2)(5)(-2)  (-2)-10 3) (m3)-1 ∙ m5  m(3)(-1)+5  m-3+5  m2 4) (9-3)2 ∙ 9-4  𝟏 𝟗 𝟏𝟎  9(-3)(2)-4  9-10

For every number a≠0 and m, n, are integers, Raising a product to powers: PROPERTIES: For every number a≠0 and m, n, are integers, (𝑎𝑏)𝑛 = 𝑎 𝑛 𝑏 𝑛 Ex: 1) (4x)3 = 43x3 = 64x3 = 𝟏 𝟑 𝟑 𝒔𝟑 = 𝟏 𝟐𝟕𝒔𝟑 2) (3s)-3 = 3-3s-3

YOU TRY IT: Simplify: (12y)-2 (-2c)5 (mz)3 ∙ m5 (9-3 n)2 ∙ 9-4

SOLUTION: No matter what integer it is, anything to the power of zero is 1. (12y)-2  𝟏 𝟏𝟐𝟐𝒚 𝟐  𝟏 𝟏𝟒𝟒𝒚 𝟐  12-2y-2 2) (-2c)5  (-2) 5c5  -32c5 3) (mz)3 ∙ m5  m3z3m5  m3+5z3  m8z3 4)(9-3z)2 ∙ 9-4  z2 𝟗 𝟏𝟎  9(-3)(2) z2 ∙ 9-4  9-10 z2

For every nonzero number a, b and integer n and m Multiplying and Scientific notation PROPERTIES: For every nonzero number a, b and integer n and m (a×10n)c(b×10m) = ac∙b×10(n)(c)+m

EXAMPLE: Simplify: (5×104)3(6×10-2 ) (3×10-5) 3(4×10-2 ) (1.13×10-7)3(9.8×105 )(3.34×1022)

SOLUTION: 1) (5×104)3(6×10-2 ) (53)(6)× 10(4)(3)-2 750× 1010 = 7.50×1012 2) (3×10-5)3(4×10-2 ) (33)(4)× 10(-5)(3)-2 108× 10-17 1.08×10-15 3) (1.13×10-7)3(9.8×105 )(3.34×1022)  (1.133)(9.8)(3.34)× 10(-7)(3)+5+22  47.23× 106  4.723× 107

𝑎 0 = 1 For every number a, Ex: 40 = 1 (-3)0 = 1 1000 = 1 ZERO: as an exponent PROPERTIES: For every number a, 𝑎 0 = 1 Ex: 40 = 1 (-3)0 = 1 1000 = 1 1,000,0000 = 1 -½ 0 =-1

𝑎 −𝑛 = 1 𝑎 𝑛 Ex: 2) (-3)-2 = 𝟏 (−𝟑)𝟐 = 𝟏 𝟗 1) 4-1 = 𝟏 𝟒 PROPERTIES: Negative numbers: as an exponents For every nonzero number a≠0, and integer n 𝑎 −𝑛 = 1 𝑎 𝑛 Ex: 2) (-3)-2 = 𝟏 (−𝟑)𝟐 = 𝟏 𝟗 1) 4-1 = 𝟏 𝟒

For every number a≠0 and m, n, are integers, Multiplying powers with same base: PROPERTIES: For every number a≠0 and m, n, are integers, 𝑎 𝑚 ∙ 𝑎 𝑛 = 𝑎 𝑚+𝑛 Ex: 1) 41∙ 43 = 41+3 = 44 = 256 2) 31 ∙ 3-3 = 31+-3 = 3-2 = 𝟏 𝟑 𝟐 = 𝟏 𝟗

For every nonzero number a, b and integer n and m Multiplying and Scientific notation PROPERTIES: For every nonzero number a, b and integer n and m (a×10n)(b×10m) = a∙b×10n+m

Raising a Power to a Power VIDEO: Raising a Power to a Power With Exponents http://www.khanacademy.org/math/algebra/exponent-equations/exponent-properties-algebra/v/exponent-properties-3

CLASSWORK: Page 436-437: Problems: As many as needed to master the concept