Chapter 8 Review Laws of

Exponents

LAW #1 Product law:

add the exponents together when multiplying the powers with the same base.

Ex:NOTE:

This operation can only be done if the base is the same!

SIMPLIFY THESE ON YOUR OWN:

€

x 3 • x 5

€

x 3y 4( ) x

2y 5( )

€

x 8

€

x 5y 9

€

33 • 312

= 33+12

= 315

LAW #2 Power of a power:

keep the base and multiply the exponents.

Ex: NOTE:

Multiply the exponents, not add them!SIMPLIFY THESE ON YOUR OWN:

€

x 3( )

5

€

x 3y 4( )

4

€

x15

€

x12y16

€

43( )

5

= 43• 5

= 415

LAW #3 Power of a product:

Distribute the power to each number or variable in the parentheses.

Ex: NOTE:

Multiply the exponents, not add them!

SIMPLIFY THESE ON YOUR OWN:

€

−2a( )2

€

3x 3y 4( )

2

€

=(−2)2a2

= 4a2

€

9x 6y 8

€

4x( )3

= 43 x 3

= 64x 3

LAW #4 Zero exponent law:

Any power raised to an exponent of zero equals one.

Ex:

SIMPLIFY THESE ON YOUR OWN:

€

x 3( )

0

€

24xy 8( )

0

€

1

€

1

€

1,283,249x 3( )

0

=1

LAW #5 Negative exponents:

To make an exponent positive, flip the base.

Ex:NOTE:

This does not change the sign of the base.

SIMPLIFY THESE ON YOUR OWN:

€

x −3

€

2x 3y 4( )

−2

€

1

x 3

€

=2−2 x −6y −8

=1

22 x 6y 8

=1

4x 6y 8

€

2−4

=1

24

=1

16

LAW #6 Quotient Property

Subtract “TOP EXPONENT MINUS THE BOTTOM EXPONENT”

Ex:NOTE:

This operation can only be done if the base is the same!

SIMPLIFY THESE ON YOUR OWN:

€

4a5

a2

€

3y 3

y 7

€

4a3

€

3y −4

€

3

y 4€

312

34

= 312−4

= 38

LAW #7 Quotient Property

Distribute the power to the top and bottom of the quotient.

Ex:

SIMPLIFY THESE ON YOUR OWN:

€

2x

3

⎛

⎝ ⎜

⎞

⎠ ⎟3

€

−2a

b

⎛

⎝ ⎜

⎞

⎠ ⎟−4

€

23 x 3

33

⎛

⎝ ⎜

⎞

⎠ ⎟

€

(−2)−4 a−4

b−4

€

8x 3

27

⎛

⎝ ⎜

⎞

⎠ ⎟

€

b4

(−2)4 a4

€

b4

16a4

€

x

2

⎛

⎝ ⎜

⎞

⎠ ⎟3

=x 3

23

=x 3

8

More examples on your own:

Ex 1:

Ex 2:

Ex 3:

Ex 4:

Ex 5:

Ex 6:

Ex 7:

Ex 8:

€

b7 • b2

€

p3( )

4

€

a2( )

3• a3

€

x 2 • xy( )2

€

4m( )2

• m3

€

3a( )3

• 2p( )2

€

82 • xy( )2

• 2x

€

q0

€

b9

€

p12

€

a9

€

x 4y 2

€

16m5

€

108a3p2

€

128x 3y 2

€

1

More examples on your own:

Ex 9:

Ex 10:

Ex 11:

Ex 12:

Ex 13:

Ex 14:

Ex 15:

Ex 16:

€

5x 2y( ) 4x 3y 2( )

€

25x 5y 4

5x 4y 2

€

x 2 • x 3

€

x13

x 5

€

55( )

2

€

5−3

€

4x 4y 2( ) 3x 2y 3

( )

€

12x 3y 4

4x 2y 2

€

20x 5y 3

€

5xy 2

€

x 5

€

x 8

€

510

= 9,765,625

€

1

53 =1

125

€

12x 6y 5

€

3xy 2

More examples on your own:

Ex 17:

Ex 18:

Ex 19:

Ex 20:

Ex 21:

Ex 22:

Ex 23:

Ex 24:

€

w3 • 3w( )4

€

p−2

€

a2b( )0

€

x −2y 3( )

−2

€

p4

p2

€

3b2

9b5

€

4x 2( )

2

4x 4

€

32 x 2y 0

x 3y −4

€

81w7

€

1

p2

€

1

€

x 4

y 6

€

p2

€

1

3b3

€

4

€

9y 4

x

More examples on your own:

Ex 25:

Ex 26:

Ex 27:

Ex 28:

Ex 29:

Ex 30:

Ex 31:

Ex 32:

€

m

n

⎛

⎝ ⎜

⎞

⎠ ⎟3

• m2n−4

€

3a2

b

⎛

⎝ ⎜

⎞

⎠ ⎟

3

•2b−1

9a

€

3

4

⎛

⎝ ⎜

⎞

⎠ ⎟3

€

2 + 3(4)2

€

m5

n7

€

6a5

b4

€

27

64

€

50

Express each number in scientific notation:

1.) 6,300 2.) 4,600,000

3.) 0.00013 4.) 0.000009

6.3 x 1034.6 x 106

1.3 x 10-4 9 x 10-6

Express each number in decimal form:

1.) 4.5 x 106 2.) 3 x 10-4

3.) 2.36 x 100 4.) 9.1 x 10-1

4,500,000 0.0003

2.36 0.91

Perform the indicated operation. Leave your answer in scientific

notation.1.) 2.)

3.) 4.)

€

2.5 x 102( ) 3 x 105

( )

€

3.4 x 10−4( ) 1.3 x 10−5

( )

€

2 x 108( )

4.5 x 105( )

€

3.6 x 10−7( )

4.8 x 10−3( )

€

7.5 x 107

€

4.42 x 10−9

€

0.4 x 103

€

4.4 x 104

€

0.75 x 10−4

€

7.5 x 10−5