1. Exponents and Division
OUR LESSON:
Exponents and Division
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2. Warm Up (a3.b4.c0) . (a6.b2) = a9b6 (-2a2b4)2 . -3a3 = -12a7b8
7ba2 . 3ab = 21a3b2
6a . 2a = 12a2
(2k) (5k3)2 = 50k7
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3. Large numbers can be represented by scientific notations
Revision
Large numbers can be represented by scientific notations
2 x 103 = 2000
So we know that 10 to the power of 3 is 10 x 10 x 10 = 1000
And 2 x 1000 = 2000
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4. 26 Exponent or Power Base Here 2 is the Base and 6 is the Power of 2
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5. Any number to the power 0 is equal to 1
26 = 2 x 2 x 2 x 2 x 2 x 2
and 20 = 1
Any number to the power 0 is equal to 1
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6. Laws of multiplying Bases
Rule for multiplying bases
am x an = a m + n
Product to a power
(zy)n = z n x y n
Power to a Power
(am)n = a m x n
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7. Lets see some examples (10ab)0 = 1 (-7xy)2 = 49x2y2
(-4)2 . (-4)3 = (-4)5 = -1024
(23ab)2 . a3 = 26 a2+3 b2 = 64 a5b2
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8. 1. Quotient Law cm ÷ cn = c m - n 2. Power of a quotient law
Lets get started
Laws of Dividing Bases
1. Quotient Law
cm ÷ cn = c m - n
2. Power of a quotient law
(z / y)n = z n / y n
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9. 4.Power to a Power (am / bm)n = amn / bmn Laws of Dividing Bases
3. Negative Exponents
X -1 = 1 / X
4.Power to a Power
(am / bm)n = amn / bmn
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10. cm ÷ cn = cm - n Quotient Law
Division of one power by another power
If c is any non-zero number and m is a larger number than n, m>n, we can write
cm ÷ cn = cm - n
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11. Base (c) is same, we only subtract the exponents
In symbols if c is any non-zero number, but if n > m, we get
Base (c) is same, we only subtract the exponents 1
cm ÷ cn = cm-n = c
n - m
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12. Lets get it better with an Example of each type
25 ÷ 23 = 2 x 2 x 2 x 2 x 2
= 25-3
= 22
2 x 2 x2
n < m
= 2 x 2 = 4
62 ÷ 65 = x 6 1 =
6 x 6 x 6 x 6 x 6
6 x 6 x 6
n > m 1 1
= 1 or =
65-2 63
216
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13. In this we raise a quotient or fraction to a power
Power of a quotient law
Dividing with the same exponents
In this we raise a quotient or fraction to a power
If z is any number, and y is any non-zero number, then
Here the power (n) is same so we multiply the z/y fraction ‘n’ number of times n n z z = = n y y
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14. Lets understand this concept with an example3 3 3 27 3 x x = = = 4 4 3 64
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15. Negative Exponent
Negative Exponent
It indicates the reciprocal of base as a fraction (not a negative number)
It indicates the reciprocal of base as a fraction (not a negative number) 1 1 1 1 -m m
X =
X =
X =
X = x x m x x -m
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16. 2 (-5) Lets try an example 1 1 23 8 = = -2 1 1 = = (-5) x (-5) 25 -3
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17. is not allowed Remember Anything to the power of zero is equal to 1
is not allowed
Anything to the power of zero is equal to 1
For example
(-7)0 = 1 1
30 = 1
= 1 30
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18. 42 . 3 63 . 3 Power to a Power 42 46 69 63 (am / bm)n = amn / bmn 3
For Example: 3 42
42 . 3 46
256 = = = 69 63
63 . 3
432
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19. c am x an = a m + n 1 cm ÷ cn = (am / bm)n = a mn / b mn Remember
When the base is same, we add /subtract the powers (exponents) as required
am x an = a m + n
When the bases are different, we multiply the powers 1
cm ÷ cn = c
m - n
(am / bm)n = a mn / b mn
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20. Your turn!
56/ = = 1/25
58/ 56 = = 25
25/ 23 = = 4
(3/5)2 = 32/52 = 9/25
= 1/72 = 1/49
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21. Questions 6. [42/33]2 = 44/36 = 256/729 7. (-3)-3 = 1 /-27 90/80 = 1
6. [42/33]2 = 44/36 = 256/729
7. (-3) = 1 /-27
90/ = 1
72/ = 49/36
[5/7]3 = 125/343
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22. Break time
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23. click here to play a game
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24. 1. (2 x 3)4 (24) (34) = (32) (25) (32) (25) = 24-5 x 34-2 = 2-1 x 32 =
9/2
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25. 2. Arrange from greatest to least
2-5, 5-2, 33 , (-4)-3 , (-3)-4 ,
2-5 = 1/32
5-2 = 1/25
33 = 27
(-4)-3= 1/64
(-3)-4 = 1/81
27, 1/25, 1/32, 1/64, 1/81
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26. 3. 3m-3 42
12 m-3-(-2) n-6 =
4m-2 n-6
= 12m-1n-6
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27. 1. Quotient Law cm ÷ cn = c m - n 2. Power of a quotient law
Lets review what we have learned today
Laws of Dividing Bases
1. Quotient Law
cm ÷ cn = c m - n
2. Power of a quotient law
(z / y)n = z n / y n
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28. 4.Power to a Power (am / bm)n = amn / bmn Laws of Dividing Bases
3. Negative Exponents
X -1 = 1 / X
4.Power to a Power
(am / bm)n = amn / bmn
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29. cm ÷ cn = cm ÷ cn = cm - n c 1 Quotient Law n - m
If c is any non-zero number and m is a larger number than n, m > n, we can write
cm ÷ cn = cm - n
In symbols if c is any non-zero number, but if n > m, we get 1
cm ÷ cn = c
n - m
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30. In this we raise a quotient or fraction to a power
Power of a quotient law
Dividing with the same exponents
In this we raise a quotient or fraction to a power
Here the power (n) is same so we multiply the z/y fraction ‘n’ number of times n n z z = n y y
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31. Negative Exponent
It indicates the reciprocal of base as a fraction (not a negative number) 1 1
X -m =
X m = x m x -m
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32. 00 Power to a Power is not allowed (am / bm)n = amn / bmn
Anything to the power of zero is equal to 1
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33. Remember to practice what you have learned!
Great Job done!
Remember to practice what you have learned!
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