10 Math Name:_______________________________

Date:________________________________

Investigating Exponent Laws

Terminology

73

Complete the following table, then answer the questions below.

Exponential Form Expanded Form Power Form Exponents

Law #1

62 x 63 (6x6)x(6x6x6) 65 2 + 3 = 5

(-5)3 x (-5)4 (-5x-5x-5)x(-5x-5x-5x-5) (-5)7 3 + 4 = 7

a x a3 (a)x(axaxa) a4 1 + 3 = 4

Law #2

37 34 = 3

7

34

3x3x3x3x3x3x3

3x3x3x3 33 7 - 4 = 3

(-4)3 (-4)2 = (-4)

3

(-4)2

-4x-4x-4

-4x-4 (-4)1 3 - 2 = 1

b5

b3

bxbxbxbxb

bxbxb b2 5 - 3 = 2

Law #3

(72)3 (7x7)x(7x7)x(7x7) 76 2 x 3 = 6

(23)3 (2x2x2)x(2x2x2)x(2x2x2) 29 3 x 3 = 9

(c4)2 (cxcxcxc)x(cxcxcxc) c8 4 x 2 = 8

What do you notice about the first three rows of the table (Law #1) ?

To multiply powers of the same base, add the exponents. bx x by = bx + y _______________________________________________________________________________________________________________________________________________

What do you notice about the second three rows of the table (Law #2) ?

To divide powers of the same base, subtract the exponents. bx by = bx - y _______________________________________________________________________________________________________________________________________________

What do you notice about the third three rows of the table (Law #3) ?

To find the power of a power, multiply the exponents. (bx)y = bxy _______________________________________________________________________________________________________________________________________________

Is it possible for a power to have a negative exponent?

Yes, because of the second law. For example: b2 b5 = b2-5 = b-3 _______________________________________________________________________________________________________________________________________________

base

exponent

power

10 Math Name:_______________________________

Date:________________________________

Investigating Exponent Laws 2

Complete the following tables – use fractions where applicable (not decimals).

Notice that each row is one multiple of the base times the row below it. Therefore, to move down the

table, it is necessary to divide by the given base.

Base 5 Base 3 Base 2

54 5x5x5x5 625 34 3x3x3x3 81 24 2x2x2x2 16

53 5x5x5 125 33 3x3x3 27 23 2x2x2 8

52 5x5 25 32 3x3 9 22 2x2 4

51 5 5 31 3 3 21 2 2

50 1 1 30 1 1 20 1 1

5-1 1

5

1

5 3-1

1

3

1

3 2-1

1

2

1

2

5-2 1

5x5

1

25 3-2

1

3x3

1

9 2-2

1

2x2

1

4

5-3 1

5x5x5

1

125 3-3

1

3x3x3

1

27 2-3

1

2x2x2

1

8

5-4 1

5x5x5x5

1

625 3-4

1

3x3x3x3

1

81 2-4

1

2x2x2x2

1

16

What do you notice about 51, 31, and 21 ?

A base with exponent one is equal to itself (the base). b1 = b _______________________________________________________________________________________________________________________________________________

What do you notice about 50, 30, and 20 ?

A base with exponent zero is equal to 1. b0 = 1 _______________________________________________________________________________________________________________________________________________

What do you notice about 5-1, 3-1, and 2-1 ?

A base with exponent negative one is equal to one over the base. b-1 = 1

b

_______________________________________________________________________________________________________________________________________________

What do you notice about 54 vs. 5-4, 32 vs. 3-2, and 23 vs. 2-3 ?

Positive and negative bases are reciprocals. b-x = 1

bx and bx =

1

b-x

_______________________________________________________________________________________________________________________________________________