numbers: exponents - university of british columbia
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Mathematics
Numbers: Exponents
Science and Mathematics
Education Research Group
Supported by UBC Teaching and Learning Enhancement Fund 2012-2014
Department of
Curr iculum and Pedagogy
F A C U L T Y O F E D U C A T I O N a place of mind
Properties of Exponents
Review of Exponent Laws
Let a and b be positive real numbers. Let x and y be real numbers.
10 a
yxyx aaa
yx
y
x
aa
a
xyyx aa
x
x
aa
1
x
x
x
b
a
b
a
xx
a
b
b
a
yxaa yx ifonly and if
Exponent Laws I
How can the following expression be simplified using
exponents?
33333
B or A Ei therE.
D.
C.
B.
A.
33
3
23
5
23
)23(
33
3
Press for hint
yxyx aaa
Solution
Answer: B
Justification: Group the expression into two separate powers of 3:
This gives the final expression 33 + 32. This cannot be simplified any
further unless the powers of three are calculated.
Answer A (35) is incorrect since 33 + 32 ≠ 35. Notice that we only add
exponents when two powers with the same base are multiplied
together, not added:
33333
33
yxyx aaa
23
yxyx aaa
3692733333
3582723D.
1255)23(C.
3692733B.
2433A.
33
33
23
5
Solution Continued
The values we are comparing in this question are small, so we can
calculate the final values of each expression:
Only answer B matches the value of the expression in the question.
The other answers give a different final value, so the expressions
are not equivalent.
Exponent Laws II
Simplify the following expression:
2
25
7
77
s impl i fied be CannotE.
D.
C.
B.
A.
23
5
5
3
77
7
17
17
Press for hint
2
2
2
5
2
25
7
7
7
7
7
77
Solution
Answer: A
Justification: This expression can be simplified in several ways:
17
77
7
)77(7
7
77
3
03
2
032
2
25
17
77
77
7
7
7
7
7
77
3
03
2225
2
2
2
5
2
25
Factor out 72 from the
numerator:
Split the fraction into the sum of
two fractions:
Exponent Laws III
Simplify the following expression: 1210
10
43
12
16
1
12
1
4
1
12
16
E.
D.
C.
B.
A.
Press for hint
xxxbaab
Solution
Answer: E
Justification: Write 1210 as a product of powers with base 3 and 4:
Alternatively, you can write the denominator as a power with base
12:
16
1
4
1
43
43
43
)43(
43
1210121210
1010
1210
10
1210
10
16
1
4
1
412
12
4)43(
12
443
12
43
122210
10
210
10
21010
10
1210
10
Exponent Laws IV
Simplify the following expression:
98
2
22
4
s impl i fied be CannotE.
D.
C.
B.
A.
9
94
54
76
2
1
22
1
22
1
22
1
Solution
Answer: B
Justification: The power of 4 can be rewritten as a power with a
base of 2:
Factor out a power of 2 from the denominator to cancel with the
numerator:
98
4
98
22
98
22
98
2
22
2
22
2
22
)2(
22
4
54544
4
494484
4
98
4
22
1
)22(2
2
2222
2
22
2
Exponent Laws V
Write the following as a single power with base 4:
0170
17
444
4
4of power s ingle a as wri tten be CannotE.
D.
C.
B.
A.
1
0
2
1
17
4
4
4
4
Solution
Answer: E
Justification: This expression
cannot be simplified any
further.
14
4
44
417
17
017
17
0
017
17
017
17
44
4
44
4
0
17
17
017
17
404
4
44
4
1. Incorrectly adding exponents
2. 40 = 1, not 0
0
17
17
17
017
17
4
4
4
4
44
4
3. Splitting the denominator
Common errors include:
Exponent Laws VI
Write the following as a single power of 2:
54
5
25.05.0
2
2of power s ingle a as wri tten be CannotE.
D.
C.
B.
A.
1
0
1
2
2
2
2
2
Solution
Answer: B
Justification: This expression can be simplified in many ways
because all the terms can be expressed as a power of 2. Two
possible solutions are shown below.
Cancel terms where possible and collect like terms:
Express all terms as a power of 2:
1545554555
4
554
5
222222)2(2
1)22(25.0
5.0
2
1
455
4
5
554
5
25.0
1
2
15.0
5.0
225.0
5.0
2
Exponent Laws VII
Which of the following powers is the largest?
34
5
2
3
4
5
6E.
5D.
4C.
3B.
2A.
435225 56
Solution
Answer: B
Justification: There are generally no rules when comparing powers
with different bases.
7776
1
6
1
255
644
813
322
5
5
2
3
4
5
6
43525 34256
Note: Large exponents tend to have
more impact on the size of a number
than large bases.
2100 1002
Exponent Laws VIII
How many of the following terms are less than 0?
23
4E.
3D.
2C.
1B.
0A.
232)3( 23
Solution
Answer: C
Justification: Simplify each term separately.
Be careful when dealing with negatives on exponents. Although every
expression has a negative sign, only –32 and –3-2 are negative. Think
about order of operations: brackets come before exponents, which
come before multiplication.
93
9
13
9
13
9)3(
2
2
2
2
99
10
9
19
)3(3033 2222
Exponent Laws IX
How many of the following are less than 1?
5E.
4D.
3C.
2B.
1A.
22222 )5.0(,)5.0(,)5.0(,2,)5.0(
Solution
Answer: D
Justification: Simplify each term separately to find the terms less
than 1.
42)5.0(
4
1
)2(
1)5.0(
4)2()5.0(
4
1
2
12
4
1
2
1)5.0(
22
2
2
22
2
2
2
2
Greater than 1
Less than 1
Less than 1
Less than 1
Less than 1
Exponent Laws X
Consider adding 3 until you obtain the value 333 as shown
below:
How many terms are there in the summation?
33
3
3
33
13
11
32
33
33
n
n
n
n
n
E.
D.
C.
B.
A.
3333...333
n terms
Solution
Answer: C
Justification: The summation is equal to 3n.
It becomes straightforward to solve for n after this step.
33
33
33
33...333
n
n terms
32
1
33
33
3
3
3
33
n
n
n
Exponent Laws XI
Let p and q be positive integers. If , which of the
following are always true?
qp
qp
qp
qp
qp
22
22
22
22
22
E.
D.
C.
B.
A.
qp
Solution
Answer: D
Justification: Rewrite the two expressions using positive exponents
of p and q:
It is now much easier to compare the two expressions.
Remember that dividing by a larger denominator gives a smaller
result.
q
q
p
p
2
12,
2
12
qp 2
1
2
1 qp 22 since
Exponent Laws XII
Let p and q be positive integers and p > q. If b > 0, find all
values of b such that
is always true.
1
1
10
10
1
b
b
b
b
b
E.
D.
C.
B.
A.
qp bb
Solution
Answer: C
Justification: Rewrite the inequality using positive exponents of p
and q:
The LHS is larger than the RHS only if bp < bq , since numbers with
smaller denominators are larger. Therefore b must be between 0 and
1 to make bp < bq (since p > q). For example, 0.52 < 0.53.
In order to choose between answers B and C, consider when b = 1.
Since we have to include the equality case, the answer is C:
qp bb
11
qp 1
1
1
1 since 1 to any power is still 1
10 b
Exponent Laws XIII
Solve for c.
xc
c
c
c
c
2E.
8D.
4C.
2B.
2
1A.
122 22
xx c
Solution
Answer: B
Justification:
2
2
2)2(
22
22
1
1
)2()2(
)2()2(
1
1
c
c
c
xx
xx
c
c
xx
xx
since xyyx aa
since if and only if yx aa yx
divide both sides by (then subtract
exponents)
x2
Exponent Laws XIV
Simplify the following:
2
1
2
199
199
1
199
199
E.
D.
C.
B.
A.
2
22
)99100(
99100
Press for hint
))((22 bababa
Difference of squares:
Solution
Answer: D
Justification: The numerator is a difference of squares:
OR
))((22 bababa
1
122199
199
1
)99100(
)99100(
)99100(
)99100)(99100(
1199199
1
)99100(
)99100(
)99100)(99100(
)99100)(99100(