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OUR LESSON:

Exponents and Division

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Warm Up

(a3.b4.c0) . (a6.b2) = a9b6

(-2a2b4)2 . -3a3 = -12a7b8

(2k) (5k3)2 = 50k7

6a . 2a = 12a2

7ba2 . 3ab = 21a3b2

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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

Revision

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26

Base

Exponent or Power

Here 2 is the Base and 6 is the Power of 2

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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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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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(10ab)0 = 1

(-4)2 . (-4)3 = (-4)5 = -1024

(-7xy)2 = 49x2y2

(23ab)2 . a3 = 26 a2+3 b2 = 64 a5b2

Lets see some examples

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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

Lets get started

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3. Negative Exponents X -1 = 1 / X

4.Power to a Power

(am / bm)n = amn / bmn

Laws of Dividing Bases

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cm ÷ cn = cm - n

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

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In symbols if c is any non-zero number, but if n > m, we get

cm ÷ cn = cm-n =

Base (c) is same, we only subtract the exponents

1c n - m

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Lets get it better with an Example of each type

25 ÷ 23 = 2 x 2 x 2 x 2 x 22 x 2 x2

= 22

62 ÷ 65 = 6 x 66 x 6 x 6 x 6 x 6

16 x 6 x 6

=

65-2

n > m

n < m

= 25-3

= 2 x 2 = 4

1or = 163

1=

216

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Power of a quotient law

In this we raise a quotient or fraction to a power

z

y

n=

zn

yn=

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

Dividing with the same exponents

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Lets understand this concept with an

example

34

3= 3 3 3

4 4 4 x x =

34 3 3

=2764

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It indicates the reciprocal of base as a fraction (not a

negative number)

X = 1x

-m

m X =1x

m

-m

It indicates the reciprocal of base as a fraction (not a

negative number)

X = 1x

X =1x

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Lets try an example

2-3

= 1

23=

1

8

(-5)-2

=1

(-5) x (-5)= 1

25

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Remember

00

is not allowed

Anything to the power of zero is equal to

1(-7)0 = 1

30

For example

1 = 1

30 = 1

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Power to a Power

(am / bm)n = amn / bmn

42

63

3= 42 . 3

63 . 3= 46

69= 256

432

For Example:

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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

cm ÷ cn =m - nc

1

(am / bm)n = a mn / b mn

Remember

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Your turn!

1. 56/ 58 = 5-2 = 1/25

2. 58/ 56 = 52 = 25

3. 25/ 23 = 22 = 4

4. (3/5)2 = 32/52 = 9/25

5. 7-2 = 1/72 = 1/49

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6. [42/33]2 = 44/36 = 256/729

7. (-3)-3 = 1 /-27

8. 90/80 = 1

9. 72/ 62 = 49/36

10. [5/7]3 = 125/343

Questions

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1. (2 x 3)4

(32) (25) =

(24) (34)

(32) (25)

= 24-5 x 34-2 = 2-1 x 32

= 9/2

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2. Arrange from greatest to least

2-5, 5-2, 33 , (-4)-3 , (-3)-4 ,

2-5 = 1/32

5-2 = 1/2533 = 27

(-4)-3= 1/64(-3)-4 = 1/81 27, 1/25, 1/32, 1/64, 1/81

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3. 3m-3 42

4m-2 n-6= 12 m-3-(-2) n-6

= 12m-1n-6

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Lets review what we have learned today

Laws of Dividing Bases

2. Power of a quotient law (z / y)n = z n / y n

1. Quotient Law cm ÷ cn = c m - n

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3. Negative Exponents X -1 = 1 / X

4.Power to a Power

(am / bm)n = amn / bmn

Laws of Dividing Bases

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cm ÷ cn = cm - n

If c is any non-zero number and m is a larger number than n, m > n, we can write

In symbols if c is any non-zero number, but if n > m, we get

cm ÷ cn = 1cn - m

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Power of a quotient law

In this we raise a quotient or fraction to a power

y

n zn

yn=

Here the power (n) is same so we multiply the z/y fraction ‘n’ number of times

Dividing with the same exponents

z

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-m

It indicates the reciprocal of base as a fraction (not a

negative number)

X -m = 1xm X m =1

x

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Power to a Power

(am / bm)n = amn / bmn

00 is not allowed

Anything to the power of zero is equal to

1

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Great Job done!

Remember to practice what you have learned!