Section 6 – 1:
Basic Exponent Rules
Selected Worked Homework Problems
The Product Rule
(AxC y D )(Bx E y F ) = A • B• x C +E • y D+F
Use the product rule to simplify each expression.
1.
( x )( x ) !
2
5
2.
Add the exponents
of the x bases
= x2+ 5
( x )( x ) !
3
3. ( 2x1 ) ( 3x 5 ) !
2
Multiply the coeffcients.
Add the exponents
of the x bases
Add the exponents
of the x bases
= x 3+ 2
!
= x7
!
= 2 • 3• x1+ 5
= x5
= 6x 6
!
4.
7.
!
( 5y )( 3y ) !
5
3
5.
( 4y )( 2y ) !
4
2
6.
( 4x )( −5x )
3
6
Multiply the coeffcients.
Add the exponents
of the y bases
!
= 5 • 3• y 5+ 3
Multiply the coeffcients.
Add the exponents
of the y bases
!
= 4 • 2• y4 + 2
Multiply the coeffcients.
Add the exponents
of the x base
= 15y 8
= 8y 6
= −20x 9
( 3y )(1y ) !
5
4
Multiply the coeffcients.
Add the exponents
of the y base
= 3•1• y 5 + 4
8.
!
= 3y 9
Chapter 6
( 3y )( y ) !
1
Multiply the coeffcients.
Add the exponents
of the x base
= 3•1• y1+ 8
9. ( 2x 3 ) ( 4x 2 )
8
!
= 3y 9
!
= 4 • −5 • x 3+ 6
Section 6 – 1 HW WKD
Multiply the coeffcients.
Add the exponents
of the x base
= 2 • 4 • x 3+ 2
= 8x 5
!
© 2016 Eitel
( )(
( )( )
)
( 2y )( 3y ) !
10. 2y 4 4x 5 !
11. 2y 4 1x 2 y 3 !
12.
Multiply the coeffcients.
There are no common bases
List the variable bases in
alphabetical order
Multiply the coeffcients.
Add the exponents
of the y bases
List the variable bases in
alphabetical order
!
Multiply the coeffcients.
Add the exponents
of the y bases
List the variable bases in
alphabetical order
= 2 •1• x 2 y 4+ 3
= 2 • 3• y 5+2
= 2x 2 y 7
= 6y 7
!
= 2 • 4 • x 5y 4
= 8x 5 y 4
(1x y )( −8y ) !
2 5
13.
4
14.
( 5y )( 4y ) !
4
5
15.
Multiply the coeffcients.
Multiply the coeffcients.
Add the exponents
of the x bases
Add the exponents
of the x bases
List the variable bases in
alphabetical order
!
List the variable bases in
alphabetical order
!
5
2
(−1x 2 y 2 )(1x 4 y 3)
Multiply the coeffcients.
Add the exponents
of the x bases
Add the exponents
of the y bases
List the variable bases in
alphabetical order
= 1• −8 • x 2 y 5+4
= 5 • 4 • y 4+5
= −1•1• x 2+4 y 2+3
= −8x 2 y 9
= 20y 9
= −x 6 y 5
(
)(
)
(
)(
)
( x1y1)( x1y1)
16. −x 5 y 3 −x 4 y 5 !
17. −3x 4 y 3 −3x 3 y 6 !
18.
Multiply the coeffcients.
Add the exponents
of the x bases
Add the exponents
of the y bases
!
List the variable bases in
alphabetical order
Multiply the coeffcients.
Add the exponents
of the x bases
Add the exponents
of the y bases
!
List the variable bases in
alphabetical order
Each variable has
an exponent of 1.
= −1• −1• x 5+4 y 3+5
= −3• −3• x 4+3 y 3+6
= x1+1 • y1+1
= x 9 y8
= 9x 7 y 9
= x 2 y2
Chapter 6
!
Section 6 – 1 HW WKD
Add the exponents
of the x bases
Add the exponents
of the y bases
!
© 2016 Eitel
19.
( x1y 3)( x1y 2 ) !
20.
( x4 y)( x3 y)
Each x variable has
an exponent of 1.
Add the exponents
of the x bases
Add the exponents !
of the y bases
Each y variable has
an exponent of 1.
Add the exponents
of the x bases
Add the exponents
of the y bases
= x1+ 1 • y 3+ 2
= x 4 + 3 • y1+ 1
= x2y5
= x7y2
The Power Rule
(A1x B y C )
D
= A1•D x B•D y C•D
Use the power rule to simplify each expression.
21.
(4 x y )
3 4 2
1
!
22.
Multiply each
exponent inside by
the exponent outside
the parentheses
!
( 2x y ) !
3 4 3
23.
Multiply each
exponent inside by
the exponent outside
the parentheses
!
( 3xy ) !
2 3
Multiply each
exponent inside by
the exponent outside
the parentheses
= 31•32x1 •2 y 2 •2
= 42 x
3•2
= 23 x
y 4•2
= 16x 6 y 8
Chapter 6
3• 3 4• 3
= 33 x 2 y 4
y
= 8x 9 y12
!
Section 6 – 1 HW WKD
= 27x 2 y 6
!
© 2016 Eitel
24.
(x y ) !
3 2 3
25.
Multiply each
exponent inside by
the exponent outside
the parentheses
= x 3 • 3 • y2 • 3
( 5xy ) !
4 2
26.
4
3
!
Place a 1 above each base
without an exponent.
Place a 1 above each base
without an exponent.
(4 x y )
(3 x y )
Multiply each
exponent inside by
Multiply each
exponent inside by
1 1 4 2
!
( 3x y )
1
the exponent outside
the parentheses
!
4 1 3
the exponent outside
the parentheses
= x 9 y6
= 5 2 • x1•2 • y 4 •2
= 31•3 • x 4 • 3 • y1 • 3
= 33 • x12 • y 3
= 25x 2 y 8
27.
( −3xy )
4 2
!
28.
= 27x12 y 3
( 6x y )
2
4 2
!
29.
( 2x y )
3 4
Place a 1 above each base
without an exponent.
Place a 1 above each base
without an exponent.
Place a 1 above each base
without an exponent.
( −3 x y )
(6 x y )
(2 x y )
Multiply each
exponent inside by
the exponent outside
the parentheses
Multiply each
exponent inside by
the exponent outside
the parentheses
Multiply each
exponent inside by
the exponent outside
the parentheses
1 1 4 2
1
!
2
4 2
1 1 3 4
!
= (−3)1•2 • x1i2 • y 4i2
= 61•2 • x 2 •2 • y 4 •2
= 21•4 • x1i4 • y 3i4
= (−3)2 • x 2 • y 8
= 62 • x 4 • y8
= 2 4 • x 4 • y12
= 9x 2 y 8
= 36x 4 y 8
= 16x 4 y12
Chapter 6
!
Section 6 – 1 HW WKD
!
© 2016 Eitel
( 9x y )
3 3 2
30.
!
( 2x y ) !
4
31.
3
32.
( 2xy )
5 2
Place a 1 above each base
without an exponent.
Place a 1 above each base
without an exponent.
Place a 1 above each base
without an exponent.
(9 x y )
(2 x y )
(2 x y )
Multiply each
exponent inside by
Multiply each
exponent inside by
Multiply each
exponent inside by
3 3 2
1
4 1 3
1
!
the exponent outside
the parentheses
1 1 5 2
the exponent outside
the parentheses
!
the exponent outside
the parentheses
= 91•2 • x 3 •2 • y 3•2
= 21•3 • x 4i3 • y1i3
= 21•2 • x1i2 • y 5i2
= 92 • x 6 • y6
= 2 3 • x12 • y 3
= 2 2 • x 2 • y10
= 81x 6 y 6
= 8x12 y 3
= 4x 2 y10
( −x y )
34. (10x 2 y 4 ) !
35.
Place a 1 above each base
without an exponent.
Place a 1 above each base
without an exponent.
Place a 1 above each base
without an exponent.
( −1 x y )
(10 x y )
(8 x y )
Multiply each
exponent inside by
the exponent outside
the parentheses
Multiply each
exponent inside by
the exponent outside
the parentheses
Multiply each
exponent inside by
the exponent outside
the parentheses
2
33.
1
3
2
!
2 1 3
1
!
2
4 2
( 8xy )
2 2
!
1 1 2 2
!
= (−1)1•3 • x 2i3 • y1i3
= 101•2 • x 2i2 • y 2i2
= 81•2 • x1i2 • y 2i2
= (−1)3 • x 6 • y 3
= 10 2 • x 4 • y 4
= 82 • x 2 • y4
= −x 6 y 3
= 100x 4 y 4
= 64x 2 y 4
Chapter 6
!
Section 6 – 1 HW WKD
!
© 2016 Eitel
( −2x y )3 !
36.
37.
Place a 1 above each base
without an exponent.
1 1 1 3
Multiply each
exponent inside by
= (−2) • x • y
3
3
2
38.
!
3 4 4
Place a 1 above each base
without an exponent.
(x y )
Multiply each
exponent inside by
Multiply each
exponent inside by
5 1 2
3 4 4
the exponent outside
the parentheses
!
the exponent outside
the parentheses
= 7 2 • x10 • y 2
3
(x y )
(7 x y )
1
= (−2)1•3 • x1i3 • y1i3
5
Place a 1 above each base
without an exponent.
( −2 x y )
the exponent outside
the parentheses
( 7x y ) !
= x12 y16
= 49x10 y 2
= −8x 3 y 3
!
The Quotient Rule
If T > B
If B > T
top exponent > bottom exponent
bottom exponent > top exponent
x T x T −B
=
1
xB
xT
1
= B−T
B
x
x
Use the Quotient Rule to simplify each expression:
41.
2x 3 y
6x 2 y 2 !
42.
2
1
reduces to
6
3
3
x
x
reduces to
2
x
1
1
y
1
reduces to
2
y
y
=
1• x •1
3•1• y
=
x
3y
Chapter 6
10xy 2
15x 3 y !
43.
14
7
reduces to
6
3
1
x
1
reduces to 2
3
x
x
1
y
1
reduces to
1
y
1
10
2
reduces to
15
3
x
1
reduces to 2
3
x
x
2
y
y
reduces to
1
y
1
!
!
=
2 •1• y
3• x 2 •1
=
2y
3x 2
14xy
6x 3 y !!
!
Section 6 – 1 HW WKD
!
=
7 •1•1
3• x 2 •1
=
7
3x 2
!
© 2016 Eitel
44.
3x 6 y 2
9x 2 y 6 !
45.
−12x 7 y 3
18x 9 y 2 !
46.
−12
−2
reduces to
18
3
7
x
1
reduces to 2
9
x
x
3
y
y
reduces to
2
y
1
3
1
reduces to
9
3
6
x
x4
reduces to
x2
1
2
y
1
reduces to 4
6
y
y
!
1• x 4 •1
=
3•1• y 4
−2 •1• y
=
3• x 2 •1
x4
= 4
3y
=
!
4xy 2
10xy 8
4
2
reduces to
10
5
1
x
1
reduces to
1
x
1
2
y
1
reduces to 6
8
y
y
!
−2y
3x 2
=
2 •1•1
5 •1• y 6
=
2
5y 5
!
2
47.
4
x y
6xy 4 !
48.
1
1
reduces to
6
6
2
x
x
reduces to
1
x
1
4
y
1
reduces to
4
y
1
−6x 3 y 2
49. 6x 3 y 8
6xy 2
9xy 2 !
−6
−1
reduces to
6
1
3
x
1
reduces to
3
x
1
2
y
1
reduces to 6
8
y
y
6
2
reduces to
9
3
1
x
1
reduces to
1
x
1
2
y
1
reduces to
1
y
1
!
!
=
1• x •1
6 •1•1
=
2 •1•1
3•1•1
=
−1•1•1
1•1• y 6
=
x
6
=
2
3
=
−1
y6
Chapter 6
!
Section 6 – 1 HW WKD
!
© 2016 Eitel
Power Rule, Product Rule, Quotient Rule
Simplify.!
!
( 3x y ) ( 2x y ) !
3 3
57.
2
58.
!
!
( 4x y )( 2x y ) !
2 3
4
60.
!
!
( 6x y ) ( x y )
2
3
3 3
!
Power Rule
Power Rule
( 3 x y )( 2x y )
( 3 x y )( 2x y )
( 27x y )( 2x y )
1i3
3
1i 3
3i3
( 4x y )( 2 x y )
( 4x y )( 2 x y )
( 4x y )( 8x y )
2 1
3
9
2 1
3
9
2 1
4 1
1i3
4 1
3
4 1
1i 3
3
6
3
y
= 4•8• x
6
3
64.
x10
1i 3
6 2
3
9
6 2
3
9
3i3
= 36 •1• x 6+3 y 2+9
= 36x 9 y11
( 3x )( 2x ) !
x
= 10
x
x8
= 10
x
3 2 3
68.
9x 9
x 9 y8
Power Rule for the top
(x
=
3i 2 i x 3+6
=
9x 9
6x 9
= 9
9x
2x 9
= 9
3x
!
Quotent Rule
!
1
= 10−8
x
(x y )
6
Product Rule
5+3
3•3 2•3
y
)
9 8
x y
x 9 y6
= 9 8
x y
Quotent Rule
Quotent Rule
2
=
3
1
x2
x 9−9
= 8−6
y
=
Chapter 6
3i2 1i2
Product Rule
1+6
y
3
Product Rule
=
4+3
= 32x 7 y 7
( x )( x ) !
63.
2
Product Rule
9+1
= 54x 5 y10
5
1i2
!
Product Rule
= 27 • 2 • x
( 6 x y )( x y )
( 6 x y )( x y )
( 36x y )( x y )
2i3
!
3+2
Power Rule
!
Section 6 – 1 HW WKD
!
1
y2
© 2016 Eitel
70.
4x 6 y 4
( x y)
3
2
!
72.
4x y
( x 3•2 y1•2
=
4x 6 y 4
x 6 y2
2
3 2
Power Rule for both factors
4
=
3
2
Power Rule for the bottom
6
( 2x y )
( 2x y )
(21i3 x 2•3 y1•3 )
(21i2 x 2•2 y 3•2 )
)
=
8x 6 y 3
4x 4 y 6
!
Quotent Rule
!
Quotent Rule
4x 9−9 y 4−2
1
=
8 i x 6−4
4 i y 6−3
4y 2
=
= 4y 2
1
=
2x 2
y3
=
Chapter 6
!
!
Section 6 – 1 HW WKD
!
© 2016 Eitel