Transcript Multiplication & Division Rule for exponents

Multiplication and Division of Exponents Notes

  

x

Multiplication Rule

n x m

x n

m

In order to use this rule the base numbers being multiplied must be the same.

x

Example:

x

3

x

4  Written in multiplication form

x

x

x

x

x

x

x

7  Using Rule

x

3  4 

x

7

   Example 1 2 3  2 5 2*2*2 2 3  5 2 8 2*2*2*2*2

  

x

3  Example 2

x

5 

x

4

x

3  5  4

x

12

  

x

 Example 3

x

4 

y

3 

y

4 

z

Remember

x

1  4

y

3  4

z x

x

1 

x

5

y

7

z

Example 4 (  2

x

3

y

5 )(3

xy

3 )(

x

2

y

)  Multiply coefficients and add exponents of like bases  6

x

(3  1  2)

y

(5  3  1)  6

x

6

y

9  

4 Example 5

x

5 (  2

x

2

y

 5

xy

2 ) In order to simplify you must distribute. Since you are multiplying when you distribute you must use the multiplication rule for exponents   8

x

5  2

y

 20

x

5  1

y

2  8

x

7

y

 20

x

6

y

2  

2.

  3, 4.

   5.

6.

1.

You Try Simplify each expression

y

4 

y

2 (  3

x

2 )(5

x

)

y

6  15

x

3

x

 3  

x

4 (

a

3

b

2 )(

a

2

b

4 )

x

7

a

5

b

6 12

y

3 (4

y

2 

y

) (

x

5

y

2 

x

4 

y

6 )

x

9

y

8  

2.

 3.

  4.

 1.

You Try Simplify (remember when adding only add coefficients of like terms) 3

x

3 (2

x

2  5

x

 2)  2

x

(4

x

 1)  4

x

(

x

 5)  (

x

2  6

x

  8) 4

x

5 (  2

x

3  6

x

)  12

x

6   6

x

5  15

x

4  6

x

3  8

x

2  2

x

4

x

2  20

x

x

2  6

x

 8 5

x

2  26

x

 8  8

x

8  24

x

6  12

x

6  8

x

8  36

x

6

  Division Rule: If the bases are the same subtract the exponents

x m x n

x m

n x x

3 5 

x

x x

 

x x

 

x

x x

x

2  OR

x

5

x

3 

x

5  3 

x

2 Always do top exponent minus bottom exponent

 2.

 3.

 1.

x

2

y

6

xy

2 6

a

7

b

3 

ab

2

x

5 

x

3

x

 Examples

x

2  1

y

6  2 

xy

4 3

a

7  1

b

3  2  3

a

6

b

Divide coefficients, subtract exponents of the like bases.

x

5  8

x

3 

x

13

x

3 

x

13  3 

x

10 Use multiplication rule on top, then use division rule 

 Special Cases ( zero power): any base raised to a power of zero equals 1

x

0  1 Here is why, when the number in the numerator is the same as the number in the denominator, the quotient is always 1.

2 4 2 4  2 * 2 * 2 * 2 2 * 2 * 2 * 2  1 So it makes sense that 2 4 2 4  2 4  4  2 0  1 

1.

 2.

 3.

a

3

b

4

a

3

b

10

x

4

y

3 

x

4

y

2

m

10  5

m

10

n

5   Examples Simplify

a

3  3

b

4  1 

a

0

b

3  1

b

3 

b

3 2

x

4  4

y

3  2  2

x

0

y

1  2 *1

y

 2

y m

10  10

n

5  5 

m

0

n

0  1*1  1