Concept Byte: Properties of Exponents

For Use With Lesson 6-1

REVIEW

Exponents indicate powers. The table below lists the properties of exponents. Assume that no denominator is equal to zero and that m and n are integers.

Example

Simplify and rewrite each expression using only positive exponents.

1. ( 5 a 3 ) ( − 3 a − 4 ) open 5 , bold italic eh cubed , close open . negative 3 bold italic eh super negative 4 end super . close

   ( 5 a 3 ) ( − 3 a − 4 ) = 5 ( − 3 ) a ( 3 + ( − 4 ) )   = − 15 a − 1   = − 15 a , or  − 15 a table with 3 rows and 2 columns , row1 column 1 , open , 5 , eh cubed , close . open . negative 3 , eh super negative 4 end super . close , column 2 equals 5 . open , negative 3 , close . eh super open . 3 plus . open , negative 4 , close . close end super , row2 column 1 , , column 2 equals negative 15 , eh super negative 1 end super , row3 column 1 , , column 2 equals , negative 15 over eh , comma or minus , 15 over eh , end table

2. ( − 4 x − 3 y 5 ) 2 open . negative 4 , bold italic x super negative 3 end super . bold italic y to the fifth . close squared

   ( − 4 x − 3 y 5 ) 2 = ( − 4 ) 2 ( x − 3 ) 2 ( y 5 ) 2   = 16 x − 6 y 10   = 16 y 10 x 6 table with 3 rows and 2 columns , row1 column 1 , open . negative 4 , x super negative 3 end super . y to the fifth . close squared , column 2 equals . open , negative 4 , close squared . open , x super negative 3 end super , close squared . open , y to the fifth , close squared , row2 column 1 , , column 2 equals 16 , x super negative 6 end super . y to the tenth , row3 column 1 , , column 2 equals . fraction 16 , y to the tenth , over x to the sixth end fraction , end table

3. 4 a b 6 c 3 a 5 b c 3 fraction 4 bold italic eh , bold italic b to the sixth , bold italic c cubed , over bold italic eh to the fifth , bold italic b , bold italic c cubed end fraction

   4 a b 6 c 3 a 5 b c 3 = 4 a ( 1 − 5 ) b ( 6 − 1 ) c ( 3 − 3 )   = 4 a − 4 b 5 c 0   = 4 b 5 a 4 table with 3 rows and 2 columns , row1 column 1 , fraction 4 eh , b to the sixth , c cubed , over eh to the fifth , b , c cubed end fraction , column 2 equals 4 . eh super open , 1 minus 5 , close end super . b super open , 6 minus 1 , close end super . c super open , 3 minus 3 , close end super , row2 column 1 , , column 2 equals 4 , eh super negative 4 end super . b to the fifth , c to the , row3 column 1 , , column 2 equals . fraction 4 , b to the fifth , over eh to the fourth end fraction , end table

Exercises

Simplify each expression. Use only positive exponents.

1. ( 2 a 3 ) ( 5 a 4 ) open 2 , eh cubed , close open 5 , eh to the fourth , close
2. ( − 3 x 2 ) ( − 4 x − 2 ) open negative 3 , x squared , close open . negative 4 x super negative 2 end super . close
3. ( 3 x 2 y 3 ) 2 open 3 , x squared , y cubed , close squared
4. ( 3 x − 4 y 3 ) 2 open . 3 x super negative 4 end super . y cubed , close squared
5. 4 a 8 2 a 4 fraction 4 , eh to the eighth , over 2 , eh to the fourth end fraction
6. 12 x 5 y 3 4 x − 1 fraction 12 , x to the fifth , y cubed , over 4 , x super negative 1 end super end fraction
7. ( 6 x 3 ) 0 3 x y 2 fraction open , 6 , x cubed , close to the , over 3 x , y squared end fraction
8. ( 2 x 4 3 ) 3 open . fraction 2 , x to the fourth , over 3 end fraction . close cubed
9. ( − 4 m 2 n 3 ) ( 2 m n ) open negative 4 , m squared , n cubed , close open 2 m n close
10. ( 2 x 3 y 7 ) − 2 open 2 , x cubed , y to the seventh , close super negative 2 end super
11. ( 3 r − 2 s 3 t 0 ) − 3 3 r s fraction open . 3 , r super negative 2 end super . s cubed , t to the . close super negative 3 end super , over 3 r s end fraction
12. ( h 7 k 3 ) 0 open , h to the seventh , k cubed , close to the
13. r 2 s 4 t 6 r 3 s 4 t − 6 fraction r squared , s to the fourth , t to the sixth , over r cubed , s to the fourth . t super negative 6 end super end fraction
14. x 2 y 4 ⋅ 16 x y fraction x squared , y , over 4 end fraction . dot , fraction 16 x , over y end fraction
15. ( s 4 t ) 2 ( s t ) open , s to the fourth , t close squared . open s t close
16. ( 1 h − 2 ) − 1 ⋅ h 3 open . fraction 1 , over h super negative 2 end super end fraction . close super negative 1 end super . dot , h cubed
17. 1 a 2 b − 3 ( a 2 b − 3 ) − 1 fraction 1 , over eh squared . b super negative 3 end super end fraction . open . eh squared . b super negative 3 end super . close super negative 1 end super
18. ( r − 1 s 2 t − 3 r − 2 s 0 t 1 ) − 1 open . fraction r super negative 1 end super . s squared . t super negative 3 end super , over r super negative 2 end super . s to the , t to the first end fraction . close super negative 1 end super
19. Reasoning Your friend tells you that ( k 2 ) − 5 = − k 10 . open , k squared , close super negative 5 end super . equals negative , k to the tenth , .  Did she apply the properties of exponents correctly? Explain why or why not.

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