Exponent Rules
1.
ChemistNATE
Simplify each of these expressions.
2 2 25=
n4 n 4=
−22 −2−5=
30 38=
x3 x2 =
2 2 2−2=
22
=
25
−22
=
−22
x2
=
x5
32
=
3−2
2−2
=
2−5
n2
=
n0
 2 2 5 =
−42 5=
 s 2 3=
 x 2 −5=
 20 5=
 x−3−4=
2.
Simplify each of these, ensuring that your answers use only positive exponents.
2 2 23
=
25 2 4
2 −3
n 
=
n 5 −4
323
=
35 34
 n 2 n 3−3
=
n8 2
 −4 
n
3.
Combine these into a single expression (get a common base)
2
1 5
32   =
3
5
8 2=
1 5
272   =
9
4.
Simplify.
a)
6-2
b)
−3
d)
()
f)
(
2
3
r 3 s4 t −5
0 −4 −4
r s t
1230
4
e)
)
−2
c)
−3
a b
a −2 b8
(-7)-2
5.
Evaluate. Write your answers as integers or fractions.
1
2
b)
c)
√5 −32
d)
6.
How is
7.
Rewrite each as a radical:
a)
a)
c)
81
8
5
3
8
1
3
125
3
√
27
1
3
−
27
1000
different from 8−3 ?
n
√m
and evaluate each expression.
b)
−
1
3
d)
125
(
2
3
64 − 23
)
125
Answers
1.
Simplify each of these expressions.
2 2 25=
27
n4 n 4=
n8
−22 −2−5=
30 38=
38
x3 x2 =
x5
2 2 2−2=
22
=
25
2-3 = (1/2)3 = 1/8
32
=
3−2
 x 2 −5=
2.
= (-2)0 = 1
4
2−2
=
2−5
210
−42 5=
3
 2 2 5 =
−22
=
−22
x-10
 20 5=
3
2
x-3 = 1/x3
n2
=
n0
n2
 x−3−4=
20
20 = 1
x2
=
x5
 s 2 3=
(-4)10
(-2)-3 = -1/8
s6
x12
Simplify each of these, ensuring that your answers use only positive exponents.
2 2 23
=
25 2 4
2 3
25 / 220 = 2-15 = 1 / 215
(3 )
=
5 4
3 3
2
2 −3
(n )
=
5 −4
(n )
n-6 / n-20 = n14
36 / 39 = 3-3 = 1 / 33
3 −3
(n n )
8 2
n
( −4 )
n
=
(n5)-3 / (n12)2 = n-15 / n24 = n-39 = 1 / n39
3.
Combine these into a single expression (get a common base)
2
5
5
2 1
3( )=
3
(23)2 × 25 = 26 × 25 = 211
8 2=
32 × 3-5 = 3-2 = 1 / 32 = 1 / 9
5
2 1
27 ( ) =
9
(33)3 × (3-2)5 = 39 × 3-10 = 3-1 = 1/3
4.
Simplify.
a)
6-2
b)
= 1 / 36
d)
−3
e)
= (3 / 2)3
= 33 / 23
= 27 / 8
f)
(
r 3 s4 t−5
0 −4 −4
r s t
4 −3
a b
a −2 b8
= a6b-11
= a6 / b11
)
c)
(-7)-2
= 1 / 72
= 1 / 49
=1
()
2
3
1230
−2
= (r3s8t-1)-2 = r-6s-16t2 = t2 / r6s16
5.
a)
Evaluate. Write your answers as integers or fractions.
1
2
81
b)
125
3
√ 125=5
= sqrt ( 81 )
=9
c)
1
3
5
√−32
d)
3
√
−
27
1000
3
= -2
6.
=
How is
8
1
3
−√ 27 −3
=
← Negative can be anywhere, really
3
√ 1000 10
different from 8−3 ?
81/3 is the CUBE ROOT of 8 … i.e. “what number gets cubed, to give 8?” … it is 2
8-3 is 1/8 CUBED. The negative flips the base ( 8 → 1/8 ) and the 3 cubes everything ... It is 1 / 512
7.
a)
Rewrite each as a radical:
8
5
3
n
√m
and evaluate each expression.
b)
= √ 8 = √ 32768=32
3
c)
−
27
5
3
1
3
√
1 1
=
=
27 3
3
125
=
2
3
= √ 125 = √ 15625=25
3
2
3
2
d)
(
64 − 3
)
125
125 3 √ 125 √ 15625 25
=(
)= 3 2= 3
=
64
√ 64 √ 4096 16
2
3
2
3