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Exponents, Squares & Square Roots
An exponent is a short way to write repeated multiplication of one factor.
Ex: 2 x 2 x 2 = 23
2 is called the base; 3 is called the exponent.
The exponent tells how many times the base is used as a factor.
Together they can be called a power. We read 23 as “two to the third power.”
Your turn: Write 5 x 5 x 5 x 5 as a power. 54
A power with exponent two is called a square. 72 can be read “seven squared” or
“seven to the second power.” Why would “square” be the name for power two??
Square Numbers
Here are the first three square numbers: 1, 4, 9, and 16. They are also called
“perfect squares.”
1 = 12
4 = 22
9 = 32
16 = 42
Complete the chart of the perfect squares with the given bases.
Base
1
2
3
4
5
6
7
Square
12 = 1
22 = 4
32 = 9
42 = 16
25
36
49
Base
8
9
10
11
12
13
14
Square
64
81
100
121
144
169
196
Base
15
16
17
18
19
20
25
Square
225
256
289
324
361
400
625
Since 32 = 9, 3 is called the square root of 9. What is the square root of 225? 15
What is the square root of 169? 13
What is the square root of 400? 20
4 2
121  11
The notation looks like this: 9  3 . Find 64  8
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Practice
1. Find the sum of the squares of the first five non-negative integers.
b) 30 02 + 12 + 22+ 32 + 42 = 30
2. Evaluate 25 – 24 + 23 - 22
b) 20
3. The sum of two consecutive even numbers is 46. What is the difference between the
squares of these two numbers? 242 - 222 = c) 92
4.
Evaluate
22009
  29  512
22000
5. Solve for x: 7x3 – 3 = 186 x3=27 so x = 3
x  9 so x = 81
x 5  4
6. Solve for x:
a) 3
d) 81
7. Evaluate 5  8  (7 +9  6)3 . (Do you know PEMDAS?) “PEMDAS” is called the Order of
Operations: Parentheses, Exponents, Multiplication & Division, Addition & Subtraction.
5 - 8 x (7 + 3/2)3
= 5 – 8 x (17/2)3
= 5 - 8(4913/8)
= 5 – 4913 = -4908
8. Evaluate 342 - 232. 1156 – 529 = 627
9. 4  44 
b) 45
10. 22009 = 22008 + ___ 22008 + 22008 = 2(22008) = 22009
d) 22008
11. If m and n are positive consecutive even integers,
find m + n if m < 2009 < n.
Solution:
2009 = 44.821… so m and n are 44 and 46. So 44+46 = 90
12. The area of a square is 225 square units.
a) What is the perimeter of the square? Sides = 15 so perim. = 4(15) = 60
b) If the square has the same perimeter as an equilateral triangle, what is the length of
the base of the equilateral triangle? 60  3  20
 410
13, Evaluate
.
47
 410
 43  64
7
4
Challenge:
14. Solve for k: 825 = 45k
2 
3 25
a) -64
  2
 
8 = 23 and 4 = 22. So and 2 2
5k
75
 210 k
b) 7.5
So 75  10k making k  7.5
612  602  412  402
15. Evaluate
3721  3600  1681  1600
d) 2
121  81  11  9  2
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