1. Properties of Integers

2. Video Complete page 33 with a partner
seventh-grade-math/exponents- powers/laws-exponents- examples/v/raising-a-number-to-the-0th- and-1st-power

3. Multiplying Numbers with Exponents
Page 34 PART A & C

4. Product of Powers
To multiply numbers with exponents that have the SAME BASE, ADD the exponents and keep the base same
Examples: 23 • 27
2 •2 • •2 •2 •2 •2 •2 •2
= 2 10
43 • 44
4 •4 • •4 •4 •4
=47
(65)( 6)
6 •6 •6 •6 •
= 66
X4 • x 6
x •x •x •x x •x •x •x •x •x
= x 10

5. 23 • 25
2 •2 • •2 •2 •2 •2 28
(x3)(x4)
x •x •x x •x •x •x x7

6. What about those with coefficients?
2x3• 4x5
= 8x8
6x4• 5x7
= 30x11
-3y4• 4y3
= -12y7

7. Power of Product
To raise a product to a power, raise each factor to the same power.
(3 · 4)2 =32 · 42 = 144
(5x)3 =52 · x2 = 25 x2

8. Power of a Power
This property is used to write an exponential expression as a single power of a base
Example: (52)3
52 • 3
52 • 52 • = 56
Example: (x2)4
x2•4
x2 •x2 •x2 •x2 = x8

9. Review
This property combines the first two multiplication properties to simplify exponential expressions
(-6 • 5) 2
(-30)2= 900
(5xy) 3
5 •5 •5 •x •x •x •y •y •y
125x3y3
(4x2)3 • x5
4 •4 •4 •x2 •x2 •x2 •x5
x •x •x • x x • x •x •x • x
64x11

10. Division of Exponents

11. Dividing Exponents
To divide with exponents that have the same base, subtract the exponents of the same numerator from the exponent of the denominator and keep the base the same
Example:
𝟑 𝟖 𝟑 𝟓 = 3· 3·3 3 ·3·3·3·3
3· 3·3 · 3 ·3
𝟑 𝟖 𝟑 𝟓 = · 3·3 3 ·3·3·3· =27
3· 3·3 · 3 ·

12. Practice
𝒙 𝟓 𝒙 𝟑
𝑥 𝑥 𝑥 𝑥 𝑥 𝑥 𝑥 𝑥
𝒙 𝟐 1
𝟓 𝟒 𝟓
𝟓 𝟑 1 =

13. Practice
3 𝟑 𝟒
1 𝟑 𝟑 = 1 27
𝟕𝒏 𝟑 𝟐𝒏 𝟓
𝟕 𝒏 𝒏 𝒏 𝟐 𝒏 𝒏 𝒏 𝒏 𝒏
𝟕 𝟐𝒏 𝟐

14. Practice 𝟖𝒎 𝟑 𝟏𝟎𝒎 𝟑 𝟖 𝒎 𝒎 𝒎 𝟏𝟎𝒎 𝒎 𝒎 8 10 = 4 5 16𝑦𝑥 𝟒 𝟗𝒙 𝟖 𝒚 𝟐
𝟖𝒎 𝟑 𝟏𝟎𝒎 𝟑
𝟖 𝒎 𝒎 𝒎 𝟏𝟎𝒎 𝒎 𝒎
8 10 = 4 5
16𝑦𝑥 𝟒 𝟗𝒙 𝟖 𝒚 𝟐
16 𝑦 𝑥 𝑥 𝑥 𝑥 9 𝒙 𝒙 𝒙 𝒙 𝒙 𝒙 𝒙 𝒙 𝒚 𝒚
16 9𝑦 𝑥 4

15. Reciprocal Review What is the reciprocal?
One of two numbers whose product is 1- also called the multiplicative inverse
Examples:
2 3 =
3 2
4 =
1 4

16. NEVER HAVE A NEGATIVE EXPONENT
Negative Exponents
NEVER HAVE A NEGATIVE EXPONENT
When dealing with negative exponents, (which is not mathematically correct) change it to a positive exponent.

17. Negative Exponent Rules
1. Put the equation into a fraction
2. in order to get rid of a negative exponent, flip the base and exponent to complete the reciprocal
3. Once flipped, the negative exponent becomes positive

18. Negative Exponents
eighth-grade-math/exponents-powers- 1/powers-negative-exponents/v/negative- exponents

19. Examples
x-3
x−3 1
1 x3
y-5
y−5 1
1 𝑦 5

20. Examples 3x-2 3𝑥−2 1 3 𝑥2 = (3x)-2 (3𝑥)−2 1 1 (3𝑥)2 = 1 9𝑥2
3 𝑥2 =
(3x)-2
(3𝑥)−2 1
1 (3𝑥)2 = 1 9𝑥2
The exponent in this example belongs only to the variable and not to the coefficient (the base of three)
Because of the parenthesis, the base of the example is everything inside the parenthesis. So, we move EVERYTHING inside the parenthesis

21. Examples:
𝑥4 1 = 𝑥4
4𝑎6 1 = 4a6

22. Example A negative coefficient is very different from a negative exponent. ONLY NEGATIVE EXPONENTS MOVE...
−5𝑥−3 𝑦−4
−5𝑦4 𝑥3