1. 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.

2. GOAL:

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

4. 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

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

6. 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

7. 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

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

9. 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

10. 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

11. 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)

12. 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

13. 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

14. 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

15. 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 = = 3-2 = 𝟏 𝟑 𝟐 = 𝟏 𝟗

16. 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

17. Raising a Power to a Power
VIDEO:
Raising a Power to a Power
With
Exponents

18. CLASSWORK: Page :
Problems: As many as needed
to master the concept