Faculty of Mathematics
Waterloo, Ontario N2L 3G1
Centre for Education in
Mathematics and Computing
Grade 6 Math Circles
November 4/5, 2014
Exponents
Quick Warm-up
Evaluate the following:
1. 4 + 4 + 4 + 4 + 4
5. 7 × 6
2. 2 + 2 + 2
6. 12 × 4
3. 9 + 9 + 9 + 9 + 9 + 9 + 9 + 9
7. 10 × 10
4. 12 + 12 + 12 + 12 + 12 + 12
8. 8 × 8
Multiplication
Where does the idea of multiplications stem from? Multiplication is simply a shorter way of
writing a repeated addition. For example 3 + 3 + 3 + 3 = 3 × 4, here we have the number 3
added with itself 4 times. So we simplify this to 3 times 4. Looking at the questions in the
quick warm up write the additions ad multiplications and the multiplications as repeated
addition.
Exponents
An exponentiation is a repeated multiplication. Similar to how a multiplication is a
repeated addition. Remember, 5 × 3 is simply 5 + 5 + 5. Similarly an exponentiation, 53 is
simply 5 × 5 × 5.
1
As shown in the picture above, we call the “lower” number the base, the “upper” number
the exponent and when refering to the base and exponent as a whole we will say the power.
When we see this notation we say “Two to the exponent three”. Note: the second and third
exponents are often referred to as squared and cubed, respectively. So we might say “two
cubed” instead of “Two to the exponent three”.
Examples:
Write the following as a multiplication then evaluate.
1. 23 = 2 × 2 × 2 = 8
2. 52
3. 34
4. 103
5. 54
Write the following numbers as exponents with the given bases
1. 32, base 2 = 25
2. 81, base 9
3. 81, base 3
4. 100, base 10
5. 1, base 4
Order of Operation: BEDMAS
If you are given 3 + 4 × 2, do you do the + or the × first? You do the × first. We have these
order of operations to make sure everyone calculates the same way. If there was no defined
order then someone could do:
3+4×2
=7×2
= 14
And someone else:
3+4×2
2
=3+8
= 11
This gives to answer for the same math problem. This is bad!!! So we have BEDMAS.What
is BEDMAS? It is a trick to remember the order of operations.
Brackets, Exponents, Division, Multiplication, Addition, Subtraction.
Example:
3 + 4 + 23 × 3
=3+4+8×3
= 3 + 4 + 24
= 31
More Examples:
• 33 − 2
• 4 ÷ 2 × 3 + 22
• (1 + 2 + 3 + 4) × 3
• (2 + 3)2 + 4
Special Cases
Base 10: what is 107 ? How about 10n ?
Base 10 powers are 1 followed by n zeros, where n is the exponent.
The first power: What is 51 ? How about 1234567891 ?
Any number raised to the exponent of 1 is equal to the base.
The power of zero: What is 80 ? How about 93847120 ?
Any non-zero number raised to the exponent of 0 is equal to 1
Now that we have covered the basics of exponents we can look at operations on exponents.
Multiplication
Since a power is simply a repeated multiplication it would only be natural to have rules for
multiplying and dividing powers.
How could we simplify :
3
2×2
This one is easy it is 22
Do you agree that the above could have been written as 21 × 21 ? What can we say about
the exponents when looking at the following equality?
21 × 21 = 22
It looks like we are adding the exponents. Consider the multiplication 25 × 23 .
25 × 23
This can be written as
2 × 2 × 2 × 2 × 2 × 23
Again as
2×2×2×2×2×2×2×2
This is 8 2’s multiplied together, it is also 28 . So we get
25 × 23 = 28
Rule:The multiplication of 2 powers with the same base is simplified to that same base
whose exponent is the sum of the 2 exponents:
am × an = a(m+n)
NOTE: THE BASES HAVE TO BE THE SAME
Examples: Simplify to a single exponent if possible
1. 53 × 59
2. 2 × 22 × 23 × 24 × 25
3. 73 × 713
4. 47 × 92
5. 24 × 42 × 24
4
division:
25
Consider 3
2
25
23
This can be written as
2×2×2×2×2
23
Again as
2×2×2×2×2
2×2×2
Now with a little work on the fraction we get
2 ×2 ×2 ×2×2
2 ×2 ×2
Here we cancel out some 2’s and we are left with two 2’s
2 × 2 = 22
25
= 22
23
Rule: The division of two powers with the same base is simplified to that same base whose
exponent is the difference of the 2 exponents:
am
= a(m−n)
n
a
Knowing this rule, can we now explain why n0 = 1? Let look at our rule but we are going
to let n = m.
am
= a(m−m)
m
a
What is any number minus that same number? It’s zero!
What is any number divided by that same number? It’s one!
this means that our equation above becomes:
a0 = 1
5
This is how we can show the property of the exponent zero.
Examples:
Simplify the following (if the bases are numbers, give their value)
1.
43
=4
42
2.
77
77
3.
3300
3298
4.
h45
h44
Power of a Power
What? (34 )5 is this even legal? Yes, and its not much more than we already covered.
Look at (34 )5 If we consider the inner exponentiation to simply be a number we can write.
34 × 34 × 34 × 34 × 34
From before we know this to be equal to 320 since 4 + 4 + 4 + 4 + 4 = 20. We can also see
this as 4 × 5 = 20. A power raised to a power is simplified by multiplying the exponents.
(an )m = an×m
Extended to more than one power, each exponent gets multiplied.
(an bk )m = an×m bk×m
Simplify:
1. (53 )4
2. (43 47 )20
3. (2 × 3 × 4)3
4. (43 )2
6
5. (347 )8
6. ((62 )2 )2
42 37
7. *** 2 3 3
(2 3 )
Negative Exponents:
25
Simplify 7
2
According to our previous rules, this gives 2−2 . What does this mean?
Looking at this as we did before we see that
25
27
This can be written as
2×2×2×2×2
2×2×2×2×2×2×2
Simplify to get
1
2×2
1
22
1
= 2−2
22
So a negative exponent in the numerator becomes a positive if it is sent to the denominator. Similarly a negative exponent in the denominator becomes a positive exponent in the
1
numerator. That is, −2 = 22
2
Examples: Simplify the following. Write the answers with positive exponents.
1.
82
84
2. 133 13−3
3.
84
8−4
4.
3−2
3−3
7
PROBLEMS
1. Write the following as exponents
(a) 4 × 4 × 4
43
(b) 7 to the 5
75
(c) 3 × 3 × 3 × 3 × 3
35
(d) 10
101
(e) 8 × 8 × 8 × 8 × 8 × 8 × 8 × 8
88
(f) 8 × 8 × 8 × 8 × 3 × 3 × 3
84 33
(g) 9 × 4 × 4 × 9 × 4
92 43
2. Evaluate the following:
(a) 106
1000000
(b) 35
243
(c) 210
1
(d) 711
71
(e) 01
0
(f) 10 + 20 + 30 + 40 + 50
5
(g) 42 + 92 − 32
88
3. What is BEDMAS and what does it stand for?
BEDMAS is a trick to remember the order of operations and it stands for: bracket,
exponent, division, multiplication, addition, subtraction
4. Evaluate
(a) 1 + 2 − 3 × 4
-9
(b) 1 − 2 × 3 + 4
-1
(c) (2 × 4 − 2)2 + 3 × 1
39
(d) (1 + 2 − 3 + 4 − 5 + 6)3
-125
5. Simplify if possible (It may help to write down the rules we covered):
(a) 22 × 22
24
8
(b) 32 × 23
32 × 23
(c) 57 × 57
51 4
(d) 64 60 60
64
(e) 43 × 65 × 42
45 65
(f) 33 3−3
30
(g) 74 77 7−9
72
6. Simplify if possible:
(a)
(b)
(c)
(d)
(e)
(f)
35
34
72
72
81
87
32
3−2
1738293
145802
6−5
6−8
31
70
8−6 =
1
86
34
1
63
7. Simplify if possible:
(a) (42 )4
48
(b) (312 )0
30 = 1
(c) ((42 )4 )2
41 6
(d) (73 72 )3
71 5
(e) (112 64 )6
1112 624
(f)
(g)
3
(34 )−1
63
(6−1 )5
35
68
8. If the population of rabbits triples every year, how many rabbits will there be in 5
years if there are currently 2?
After one year 2 × 3, after 2 years 2 × 3 × 3, after 3 years 2 × 3 × 3 × 3, continuing we
get 2 × 35 = 486
9
9. If a bacteria population starts at 100 quadruples every hour, how many bacteria will
there be in 6 hours?
100 × 46 = 100 × 4096 = 409600
10. The memory capacity of a computer doubles every year. If you can store 1000 songs
on your MP3 player now, how many songs will you be able to store in 10 years?
1000 × 210 = 1000 × 1024 = 1024000
11. You go back in time and tell you parents to buy into Apple. Your parents wisely listen
you and invest $1000. Since then, The value of apple has tripled 4 times. how much
would your parents $1000 investment be worth now?
1000 tripled 4 times means 3 × 3 × 3 × 3 = 34 so your parents 1000 dollar investment
would be worth 1000 × 34 = 81000
12. ** Express 817 as a power of base 3.
If we look at 81 we notice we can write it as 34 since 34 = 81
So we can write the above as (34 )7
From here we can multiply 4 by 7 to get the answer
327
164 × 643
as a power of base 2.
13. ** Express
224
We can simplify this similar to question 11. Here we will change every thing to a power
of base 2 then use the rules we learned.
(24 )4 × (26 )3
224
216 × 218
224
10
2
14. **
3
27
(23 35 4−3 )−2
23 3−11 45
15. ** If you have 0 < 10n < 1 000 000 000, What is the max value of 3−n ?
The max value is when n = 0, and therefore 3−n = 1
10