Operations With Exponents

An exponent indicates how many times a number is used as a factor.

Example 1

Write using exponents.

1. 3 · 3 · 3 · 3 · 3 = 3 5 3 middle dot 3 middle dot 3 middle dot 3 middle dot 3 equals , 3 to the fifth
2. a · a · b · b · b · b = a 2 b 4 eh middle dot , eh middle dot b middle dot b middle dot b middle dot b equals , eh squared , b to the fourth

The patterns shown below indicate that a 0 = 1 eh to the , equals 1 and that a − n = 1 a n . eh super negative n end super , equals , fraction 1 , over eh to the n end fraction , .

Example 2

Write each expression so that all exponents are positive.

1. a − 2 b 3 = 1 a 2 ⋅ b 3 = b 3 a 2 eh super negative 2 end super . b cubed , equals , fraction 1 , over eh squared end fraction , dot , b cubed , equals . fraction b cubed , over eh squared end fraction
2. x 3 y 0 z − 1 = x 3 ⋅ 1 ⋅ 1 z = x 2 z x cubed , y to the . z super negative 1 end super , equals , x cubed , dot 1 dot , 1 over z , equals , fraction x squared , over z end fraction

You can simplify expressions that contain powers with the same base.

Example 3

Simplify each expression.

1. a. b 5 · b 3 = b 5 + 3 Add exponents to multiply   = b 8 powers with the same base. table with 2 rows and 4 columns , row1 column 1 , b to the fifth , middle dot , b cubed , column 2 equals , column 3 b super 5 plus 3 end super , column 4 cap add exponents to multiply , row2 column 1 , , column 2 equals , column 3 b to the eighth , column 4 powers with the same base. , end table
2. b. x 5 x 7 = x 5 − 7 Subtract exponents to divide   = x − 2 = 1 x 2 powers with the same base . table with 2 rows and 3 columns , row1 column 1 , fraction x to the fifth , over x to the seventh end fraction , column 2 equals . x super 5 minus 7 end super , column 3 cap subtractexponentstodivide , row2 column 1 , , column 2 equals , x super negative 2 end super , equals , fraction 1 , over x squared end fraction , column 3 powerswiththesamebase . . , end table

You can simplify expressions that contain parentheses and exponents.

Example 4

Simplify each expression.

1. ( a b n ) 3 = a 3 b 3 n 3 Raise each factor in the parentheses to the third power . open , fraction eh b , over n end fraction , close cubed . equals . fraction eh cubed , b cubed , over n cubed end fraction . table with 2 rows and 1 column , row1 column 1 , cap raiseeachfactorinthe , row2 column 1 , parenthesestothethirdpower . . , end table

2. ( c 2 ) 4 = c 2 · 4 = c 8 open , c squared , close to the fourth . equals . c super 2 middle dot 4 end super . equals , c to the eighth

   Multiply exponents to raise a power to a power.

Exercises

Write each expression using exponents.

1. x · x · x x middle dot , x middle dot x
2. x · x · x · y · y x middle dot , x middle dot x middle dot y middle dot y
3. a · a · a · a · b eh middle dot , eh middle dot eh middle dot eh middle dot b
4. a ⋅ a ⋅ a ⋅ a b ⋅ b fraction eh dot eh dot eh dot eh , over b dot b end fraction

Write each expression so that all exponents are positive.

5. c − 4 c super negative 4 end super
6. m − 2 n 0 m super negative 2 end super . n to the
7. x 5 y − 7 z − 3 x to the fifth . y super negative 7 end super . z super negative 3 end super
8. a b − 1 c 2 eh b super negative 1 end super . c squared

Simplify each expression. Use positive exponents.

9. d 2 d 6 d squared , d to the sixth
10. a 5 a 2 fraction eh to the fifth , over eh squared end fraction
11. c 7 c fraction c to the seventh , over c end fraction
12. n 3 n 6 fraction n cubed , over n to the sixth end fraction
13. a 5 b 3 a b 8 fraction eh to the fifth , b cubed , over eh , b to the eighth end fraction
14. ( 3 x ) 2 open 3 x close squared
15. ( a b ) 4 open , eh over b , close to the fourth
16. ( x z y ) 6 open , fraction x z , over y end fraction , close to the sixth
17. ( c 3 ) 4 open , c cubed , close to the fourth
18. ( x 2 y 5 ) 3 open . fraction x squared , over y to the fifth end fraction . close cubed
19. ( u 4 v 2 ) 3 open , u to the fourth , v squared , close cubed
20. ( p 5 ) − 2 open , p to the fifth , close super negative 2 end super
21. ( 2 a 4 ) ( 3 a 2 ) 6 a 3 fraction open , 2 , eh to the fourth , close . open , 3 , eh squared , close , over 6 , eh cubed end fraction
22. ( x − 2 ) 3 open , x super negative 2 end super , close cubed
23. ( m g 3 ) − 1 open , m g cubed , close super negative 1 end super
24. g − 3 g − 1 g super negative 3 end super . g super negative 1 end super
25. ( 3 a 3 ) 2 18 a fraction open , 3 , eh cubed , close squared , over 18 eh end fraction
26. c 3 d 7 c − 3 d − 1 fraction c cubed , d to the seventh , over c super negative 3 end super . d super negative 1 end super end fraction

Table of Contents