Section P.2: Properties of Negative and Zero Exponents
Chapter P – Polynomials
#1-42: Simplify
1) a)
𝑥2
𝑥2
Anything divided by itself is equal to 1. You may also subtract the exponents to get x0
Solution: 1
b) x0
If you solved part (a) by subtracting the exponents you would get x2-2 which is equal to x0. x0 must equal
to the answer to part a.
Solution: 1
3) a)
34
34
Anything divided by itself is equal to 1. You may also subtract the exponents to get 30
Solution: 1
b) 30
If you solved part (a) by subtracting the exponents you would get 34-4 which is equal to 30. 30 must
equal to the answer to part a.
Solution: 1
5) a) 2*30
You can check this on your calculator since it only has numbers. By hand the 30 is equal to 1.
= 2*1
Solution: 2
b) (2*3)0
Again, you can check this on your calculator.
The 2 is in the parenthesis, and this really equals 20*30 which is 1*1
Solution 1
7) a) -1*30
If a number is not in a parenthesis, it doesn’t turn to 1 when you apply the 0 exponent rule.
= -1*1 (only the 30 turns to a 1)
Solution: -1
b) (-1*3)0
Since the -1 is in a parenthesis, the entire -1*3 turns to a 1.
Solution: 1
9) a) -30
If a negative sign is not in a parenthesis, it doesn’t turn to 1 when you apply the 0 exponent rule.
Think of this as = -1*30 (only the 30 turns to a 1)
= -1*1
Solution: -1
b) (-3)0
everything is in a parenthesis, this will equal 1.
Solution: 1
11) a) -1*y0
= -1 * 1 (only the y0 turns into a 1)
Solution: -1
b) (-1*y)0
The negative is in a parenthesis, so the 0 exponent applies to both the -1 and the y. This is equal to 1.
Solution: 1
13) a) -y0
This has the same meaning as problem 11a) that is it equals -1*y0
= -1*y0
= - 1 *1
Solution: -1
b) (-y)0
Anytime a 0 exponent applies to the contents of a parenthesis, the contents of the parenthesis turns
into 1 when you apply the exponent.
Solution: 1
15) a) 5*60
= 5*1
Solution: 5
b) (5*6)0
everything is in a parenthesis, this will equal 1.
Solution: 1
17) a) 2x0
=2*1
Solution: 2
b) (2x)0
everything is in a parenthesis, this will equal 1.
Solution: 1
19) a) 3xy0
Only the y0 turns into a 1
= 3x*1
Solution: 3x
b) (3xy)0
everything is in a parenthesis, this will equal 1
Solution: 1
21) a) 3x0y
= 3 * x0 * y
Only the x0 turns into a 1.
=3*1*y
Solution: 3y
b) (3x0y)0
Since the entire 3x0y is in the parenthesis, it all turns to 1 when the 0 exponent rule is applied.
Solution: 1
23) x0
Solution: 1
25) 30
Solution: 1
27) -30
I can simplify this using my calculator. By hand, I need to think of this as -1*30, and only the 30 turns into
a 1.
= -1*30
= -1*1
Solution: -1
29) (-2)0
I can simplify this using my calculator. By hand, since the entire -2 is in a parenthesis the entire problem
simplifies to 1.
Solution: 1
31) -w0
Think of this as -1*w0 = -1*1
Solution: -1
33) (-w)0
The 0 exponent applies to the content of a parenthesis and the parenthesis simplifies to 1 when you
apply the exponent.
Solution: 1
35) 2c0
= 2 * c0 (only the c0 turns into a 1)
= 2*1
Solution: 2
37) (2x)0
The 0 exponent applies to the content of a parenthesis and the parenthesis simplifies to 1 when you
apply the exponent.
Solution: 1
39) 2bc0
= 2b*1
Solution: 2b
41) (2xy0)0
The 0 exponent applies to the content of a parenthesis and the parenthesis simplifies to 1 when you
apply the exponent.
Solution: 1
43) a) 3−2
This can be done using a calculator. By hand I need to make the exponent positive by creating a fraction
with a 1 in the numerator.
1
= 32
Solution:
𝟏
𝟗
b) x-2
I need to make the exponent positive by creating a fraction with a 1 in the numerator.
Solution:
𝟏
𝒙𝟐
45) a) 3-3
1
= 33
𝟏
Solution: 𝟐𝟕
b) z-3
𝟏
Solution: 𝒛𝟑
47) 4-3
1
= 43
𝟏
Solution: 𝟔𝟒
49) x-7
Solution:
51) a)
𝟏
𝒙𝟕
1
3−2
This can be done using a calculator. By hand I need to make the exponent positive by moving the 3-2 up
to the numerator.
= 1*32
Now order of operations requires me to do the 32 first.
= 1*9
Solution: 9
b)
1
𝑥 −2
I need to make the exponent positive by moving the x-2 up to the numerator.
Solution: 1x2 or just x2
53) a)
2
3−2
This can be done using a calculator. By hand I need to make the exponent positive by moving the 3-2 up
to the numerator.
= 2*32
Now order of operations requires me to do the 32 first.
= 2*9
Solution: 18
b)
2
𝑥 −2
I need to make the exponent positive by moving the x-2 up to the numerator.
Solution: 2x2
55)
7
5−1
= 7*51
= 7*5
Solution: 35
57)
2
𝑥 −5
Solution: 2x5
59)
𝑥3
𝑥 −6
I need to make all of the exponents positive. I can do this by moving the x-6 up to the numerator. My
answer will not be a fraction as I will have taken the only thing out of the denominator away from the
denominator.
= x3*x6
Solution: x9
61)
𝑥2
𝑥 −6
Move the x-6 up
= x2x6
Solution: x8
63)
𝑥3
𝑥 −5
= x3x5
Solution: x8
65) a)
3−2
2−3
I can simplify this on my calculator. By hand move the 3-2 down to the denominator, and the 2-3 up to
the numerator.
23
= 32
𝟖
Solution: 𝟗
b)
𝑥 −2
𝑦 −3
move the x down and the y up and make the exponents positive.
Solution:
67)
=
𝒚𝟑
𝒙𝟐
2−3
5−1
51
23
Solution:
69)
𝟓
𝟖
𝑥 −3
𝑦 −1
By hand move the x-3 down to the denominator, and the y-1 up to the numerator.
Solution:
71)
=
𝒚
𝒙𝟑
𝑥 −3
𝑥 −1
𝑥1
𝑥3
(𝐼 𝑐𝑎𝑛 𝑡ℎ𝑖𝑘 𝑜𝑓 𝑡ℎ𝑖𝑠 𝑎𝑠:
𝑥
.
𝑥𝑥𝑥
𝐶𝑎𝑛𝑐𝑒𝑙 𝑡ℎ𝑒 𝑥 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 𝑤𝑖𝑡ℎ 𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑥 ′ 𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟)
Really we should just subtract the exponents and leave the two x’s in the denominator.
𝟏
Solution: 𝒙𝟐
73)
𝑥 −5
𝑥 −1
𝑥1
= 𝑥5
(𝐼 𝑐𝑎𝑛 𝑡ℎ𝑖𝑘 𝑜𝑓 𝑡ℎ𝑖𝑠 𝑎𝑠:
𝑥
. 𝐶𝑎𝑛𝑐𝑒𝑙 𝑡ℎ𝑒 𝑥 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 𝑤𝑖𝑡ℎ 𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑥 ′ 𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟)
𝑥𝑥𝑥𝑥𝑥
Really we should just subtract the exponents and leave the four x’s in the denominator.
𝟏
Solution: 𝒙𝟒
75) a)
3−2
33
I need to move the 3-2 down to the denominator. I will leave a 1 in the numerator as I can’t write a
fraction without a numerator.
1
= 33 ∗32 now I will add the exponents
1
= 35 I will use my calculator to simplify 35
Solution:
𝟏
𝟐𝟒𝟑
𝑥 −2
𝑥6
b)
I need to move the x-2 down to the denominator. I will leave a 1 in the numerator as I can’t write a
fraction without a numerator.
1
= 𝑥6𝑥2
Solution:
77)
𝟏
𝒙𝟖
𝑥 −2
𝑥5
I need to move the x-2 down to the denominator. I will leave a 1 in the numerator as I can’t write a
fraction without a numerator.
=
1
𝑥2𝑥5
𝟏
Solution: 𝒙𝟕
79) a)
2∗3−2
5−2
I will move the 3-2 down and the 5-2 up.
=
2∗52
32
=
2∗25
9
Solution:
b)
𝟓𝟎
𝟗
4𝑥 −2
𝑦 −3
I will move the x-2 down and the y-3 up.
Solution:
𝟒𝒚𝟑
𝒙𝟐
3𝑥 −1
𝑦 −3
81)
Solution:
3𝑥 −1
𝑥 −3
83)
=
𝟑𝒚𝟑
𝒙
3𝑥 3
𝑥1
now subtract the exponents, leave the x in the numerator and my answer will not be a fraction.
The bigger exponent is in the numerator, so the answer is not a fraction.
Solution: 3x2
85) a)
3∗22
5−2
I will move the 5-2 up to the numerator. I won’t need to write a fraction since I have taken everything
away from the denominator.
= 3*22*52
=3*4*25
Solution: 300
b)
4𝑥 2
𝑦 −3
Solution: 4x2y3
87) a)
3∗2−2
52
I will move the 2-2 down to the denominator. Nothing else has a negative exponent. Nothing else
moves.
3
= 52 ∗22
3
= 25∗4
Solution:
b)
𝟑
𝟏𝟎𝟎
4𝑥 −2
𝑦3
I will move the x-2 down to the denominator. Nothing else has a negative exponent. Nothing else moves.
Solution:
𝟒
𝒙𝟐 𝒚𝟑
89)
2𝑥 −3
𝑦
Move the x down leave the rest alone. The only negative exponent is the one that applies to the x.
𝟐
Solution: 𝒙𝟑𝒚
91)
2𝑥 3
𝑦 −5
Move the y up, leave everything else fixed. My answer should not be a fraction as I moved the entire
denominator up to the numerator.
Solution: 2x3y5
93)
6𝑥 −2 𝑦 −6
21𝑥 5 𝑦 −4
Move the x-2, y-6 and y-4
6𝑦 4
= 21𝑥 5 𝑥2 𝑦6
6
Reduce the 21, then subtract the exponents on the letters. The y winds up in the denominator since the
bigger exponent of the y’s is in the denominator. The x winds up in the denominator since both x’s are in
the denominator. Add the exponents of the x’s since they are on the same level of the fraction.
Solution:
95)
𝟐
𝟕𝒙𝟕 𝒚𝟐
9𝑥 2 𝑦 6
30𝑥 −5 𝑦 −4
Only move the x-5 and y-4
=
9𝑥 2 𝑥 5 𝑦 6 𝑦 4
30
Reduce the
Solution:
97)
9
and
30
𝟑𝒙𝟕 𝒚𝟏𝟎
𝟏𝟎
2𝑥 −3
𝑥
2
= 𝑥𝑥 3
𝟐
Solution: 𝒙𝟒
add the exponents of the x and y.
99)
2𝑥 3
𝑥 −5
= 2x3x5
Solution: 2x8
101) a)
23
25
Subtract the exponents, leave the 2 in the denominator as that is where the larger exponent is. I have
to leave 1 in the numerator.
1
= 22
Solution:
b)
𝟏
𝟒
𝑥3
𝑥5
Subtract the exponents, leave the x in the denominator as that is where the larger exponent is. I have to
leave 1 in the numerator.
1
= 𝑥 5−3
Solution:
103)
𝟏
𝒙𝟐
𝑦2
𝑦6
Subtract the exponents. The y will end up in the denominator as the larger exponent is in the
denominator.
𝟏
Solution: 𝒚𝟒
105)
𝑥𝑦 3
𝑥2𝑦
Subtract the exponents of the x and y. The x winds up in the denominator, the y in the numerator.
When you divide with positive exponents you subtract the exponents and leave the letter where the
larger exponent starts.
𝑦 3−1
= 𝑥 2−1
Solution:
𝒚𝟐
𝒙
107) a) 2*3-2
Create a fraction and move the 3-2 to the denominator. Leave the 2 in the numerator since it does not
have a negative exponent.
2
= 32
Solution:
𝟐
𝟗
b) 2x-2
Create a fraction and move the x-2 to the denominator. Leave the 2 in the numerator since it does not
have a negative exponent.
Solution:
𝟐
𝒙𝟐
109) 5x-3
Create a fraction and move the x-3 to the denominator. Leave the 5 in the numerator since it does not
have a negative exponent.
Solution:
𝟓
𝒙𝟑
111) 2x-1
Create a fraction and move the x-1 to the denominator. Leave the 2 in the numerator since it does not
have a negative exponent.
Solution:
𝟐
𝒙
113) x3x-4
Create a fraction and move the x-4 to the denominator. Leave the x3 in the numerator since it does not
have a negative exponent.
𝑥3
𝑥4
now subtract the exponents and leave the x in the denominator
Solution:
𝟏
𝒙
115) y-2y3
Move the y-2 down, but don’t move the y3
𝑦3
= 𝑦2 now subtract exponents.
Solution: y
117) a) 3*4-1*5
Move the 4-1, but leave the 3 and the 5.
=
3∗5
41
Solution:
𝟏𝟓
𝟒
b) 2𝑦 −5x
Move the y-5, but leave the 2 and the x.
Solution:
𝟐𝒙
𝒚𝟓
119) a) 3*2*5-2
Move the 5-2, but leave the 3 and the 2.
3∗2
= 52
Solution:
𝟔
𝟐𝟓
b) 4xy-3
Move the y-3, but leave the 4 and the x.
Solution:
𝟒𝒙
𝒚𝟑
121) a) 3-1*4*5-1
Move both the 3-1 and 5-1, leave the 4.
=
4
31 ∗51
=
4
3∗5
Solution:
𝟒
𝟏𝟓
b) 4-1xy-3
Move both the 4-1 and y-3, leave the x.
=
𝑥
41 𝑦3
𝒙
Solution: 𝟒𝒚𝟑
123) 6xy-4
Only move the y down.
𝟔𝒙
Solution: 𝒚𝟒
125) 6-1xy-3
Move both the 6-1 and y-3, leave the x.
𝑥
= 61 𝑦 3
Solution:
𝒙
𝟔𝒚𝟑
127) 6x-2y-3
Leave the 6, but move the x and y
𝟔
Solution: 𝒙𝟐𝒚𝟑
129) 2-1x-2y-3
Everything moves to the denominator. I will have to write a 1 in the numerator as I can’t write a fraction
without a numerator.
1
= 21 𝑥 2 𝑦 3
𝟏
Solution: 𝟐𝒙𝟐𝒚𝟑
131) 6x-2x-3
=
6
𝑥2𝑥3
𝟔
Solution: 𝒙𝟓
133) 2-1x-2x-4
Move everything to the denominator. I will need to leave a 1 in the numerator.
1
= 21 𝑥 2 𝑥 4
Now, add the exponents of the x’s
Solution:
𝟏
𝟐𝒙𝟔
2 −4
3
135) a) ( )
Take the reciprocal of the fraction and make the exponent positive.
3 4
= (2)
Now raise the 3 and 2 to the 4th power.
34
= 24
𝟖𝟏
Solution: 𝟏𝟔
𝑥 −4
b) (𝑦)
Take the reciprocal of the fraction and make the exponent positive.
𝑦 4
𝑥
( ) I need to multiply the exponents by 4. I will write the exponents in, but it is not necessary.
4
𝑦1
= (𝑥 1 )
𝑦 1∗4
=𝑥 1∗4
𝒚𝟒
Solution: 𝒙𝟒
2
−2
137) 𝑎) (3∗52 )
Take the reciprocal of the fraction and make the exponent positive.
2
3∗52
)
2
=(
I can simplify the inside of the parenthesis next.
75 2
2
=( )
Now I need to square each number
𝟓𝟔𝟐𝟓
𝟒
Solution:
𝑥
−2
b) (3𝑦2 )
Take the reciprocal of the fraction and make the exponent positive.
2
3𝑦 2
)
𝑥
=(
=
32 𝑦 2∗2
𝑥 1∗2
Solution:
𝟗𝒚𝟒
𝒙𝟐
−2
3∗4−1
)
5
139) a) (
I can simplify the inside of the parenthesis first.
−2
3
= (5∗41 )
3 −2
= (20)
Now I can take the reciprocal to make the exponent positive.
20 2
=( 3 )
=
202
32
𝟒𝟎𝟎
𝟗
Solution:
−2
3∗𝑥 −1
)
5
b) (
I can simplify the inside of the parenthesis first.
3
−2
= (5𝑥)
Now I can take the reciprocal to make the exponent positive.
5𝑥 2
3
= ( )
=
52 𝑥 1∗2
32
𝟐𝟓𝒙𝟐
𝟗
Solution:
−1
3𝑥 −3
)
𝑦
141) (
I can simplify the inside of the parenthesis first.
3 −1
)
𝑥3𝑦
=(
Now I can take the reciprocal to make the exponent positive.
1
𝑥3𝑦
)
3
=(
Solution:
𝒙𝟑 𝒚
𝟑
2𝑥 −2
143) (𝑦−4 )
Simplify the inside of the parenthesis first.
= (2𝑥𝑦 4 )−2
Now I can take the reciprocal to make the exponent positive.
1
= (2𝑥𝑦4 )2
1
= 22 𝑥 1∗2 𝑦4∗2
Solution:
𝟏
𝟒𝒙𝟐 𝒚𝟖
2
3𝑥 −3
)
𝑦
145) (
Simplify the inside of the parenthesis first.
2
3
= (𝑥 3 𝑦) no need to flip this as there are no more negative exponents left. Just square the 3 and
multiply the exponents by 2.
Solution:
𝟗
𝒙𝟔 𝒚𝟐
2𝑥 3
)
𝑦 −4
147) (
Simplify the inside of the parenthesis first.
= (2𝑥𝑦 4 )3 no need to flip this as there are no more negative exponents left. Just cube the 2 and
multiply the exponents by 3.
Solution: 8x3y12