1. The mathematician’s shorthand
Exponents
The mathematician’s shorthand

2. Is there a simpler way to write 5 + 5 + 5 + 5? 4 · 5
Just as repeated addition can be simplified by multiplication, repeated multiplication can be simplified by using exponents. For example:
2 · 2 · 2 is the same as 2³, since there are three 2’s being multiplied together.

3. Exponential form – a number written with a base and an exponent. (23)
Likewise, 5 · 5 · 5 · 5 = 54, because there are four 5’s being multiplied together.
Power – a number produced by raising a base to an exponent. (the term 27 is called a power.)
Exponential form – a number written with a base and an exponent. (23)
Exponent – the number that indicates how many times the base is used as a factor. (27)
Base – when a number is being raised to a power, the number being used as a factor. (27)

4. Evaluating exponents is the second step in the order of operations
Evaluating exponents is the second step in the order of operations. The sign rules for multiplication still apply.

5. Writing exponents
3 · 3 · 3 · 3 · 3 · 3 = 36 How many times is 3 used as a factor?
(-2)(-2)(-2)(-2) = (-2)4 How many times is -2 used as a factor?
x · x · x · x · x = x5 How many times is x used as a factor?
12 = 121 How many times is 12 used as a factor?
36 is read as “3 to the 6th power.”

6. Evaluating Powers 26 = 2 · 2 · 2 · 2 · 2 · 2 = 64 83 = 8 · 8 · 8 = 512
54 = 5 · 5 · 5 · 5 = 625
Always use parentheses to raise a negative number to a power.
(-8)2 = (-8)(-8) = 64
(-5)3 = (-5)(-5)(-5) = -125
(-3)5 = (-3)(-3)(-3)(-3)(-3) = -243

7. (-3)(-3)(-3)(-3)(-3)(-3) = (-3)6 = 729
When we multiply negative numbers together, we must use parentheses to switch to exponent notation.
(-3)(-3)(-3)(-3)(-3)(-3) = (-3)6 = 729
You must be careful with negative signs!
(-3)6 and -36 mean something entirely different.

8. Note: When dealing with negative numbers,
Note: When dealing with negative numbers, *if the exponent is an even number the answer will be positive. (-3)(-3)(-3)(-3) = (-3)4 = 81 *if the exponent is an odd number the answer will be negative. (-3)(-3)(-3)(-3)(-3) = (-3)5 = -243

9. In general, the format for using exponents is: (base)exponent where the exponent tells you how many times the base is being multiplied together. Just a note about zero exponents: powers such as 20, 80 are all equal to 1. You will learn more about zero powers in properties of exponents and algebra.

10. Simplifying Expressions Containing Powers
50 – 2(3 · 23)
= 50 – 2(3 · 8) Evaluate the exponent.
= 50 – 2(24) Multiply inside parentheses.
= 50 – 48 Multiply from left to right.
= 2 Subtract from left to right.

11. Simplify and Solve (3 - 62) =
42 + (3 · 42)
27 + (2 · 52)
(-3)5
2( )
A population of bacteria doubles in size every minute. The number of bacteria after 5 minutes is 15(25). How many bacteria are there after 5 minutes?

12. Properties of Exponents
Multiplying, dividing powers and zero power.

13. The factors of a power, such as 74, can be grouped in different ways
The factors of a power, such as 74, can be grouped in different ways. Notice the relationship of the exponents in each product.
7 · 7 · 7 · 7 = 74
(7 · 7 · 7) · 7 = 73 · 71 = 74
(7 · 7) · (7 · 7) = 72 · 72 = 74

14. Multiplying Powers with the Same Base
To multiply powers with the same base, keep the base and add the exponents.
35 · 38 = 35+8 = 313
am · an = a m+n

15. Multiply
35 · 32 = 35+2 = 37
a10 · a10 = a10+10 = a20
16 · 167 = = 168
64 · 44 = Cannot combine; the bases are not the same.

16. Dividing Powers with the Same Base
To divide powers with the same base, keep the base and subtract the exponents.
69 = 69-4 = 65 64
bm = bm-n bn

17. Divide
1009 = = 1006
1003
x8 = Cannot combine; the bases are not the same. y5
When the numerator and denominator of a fraction have the same base and exponent, subtracting the exponents results in a 0 exponent.
1 = 42 = 42-2 = 40 = 1 42

18. The zero power of any number except 0 equals 1
The zero power of any number except 0 equals = 1 (-7)0 = 1 a0 = 1 if a ≠ 0

19. How much is a googol? 10100 Life comes at you fast, doesn’t it?

20. Extremely small numbers
Negative Exponents
Extremely small numbers

21. Negative exponents have a special meaning. The rule is as follows:
Basenegative exponent =
Base1/positive exponent
4-1 = 1 41

22. Look for a pattern in the table below to extend what you know about exponents. Start with what you know about positive and zero exponents.
103 = 10 · 10 · 10 = 1000
102 = 10 · 10 = 100
101 = 10 = 10
100 = 1 = 1
10-1 = 1/10
10-2 = 1/10 · 10 = 1/100
10-3 = 1/10 · 10 · 10 = 1/1000

23. Example:
10-5 = 1/105 = 1/10·10·10·10·10 = 1/100,000 =
So how long is 10-5 meters?
10-5 = 1/100,000 = “one hundred-thousandth of a meter.
Negative exponent – a power with a negative exponent equals 1 ÷ that power with a positive exponent.
5-3 = 1/53 = 1/5·5·5 = 1/125

24. Evaluating negative exponents
(-2)-3 = 1/(-2)3 = 1/(-2)(-2)(-2) = -1/8
5-3 = 1/53 = 1/(5)(5)(5) = 1/ 125
(-10)-3 = 1/(-10)3 = 1/(-10)(-10)(-10) = -1/1000 =
3-4 · 35 = = 31 = 3 Remember Properties of Exponents: multiply same base you keep the base and add the exponents.

25. Evaluate exponents: Get your pencil and calculator ready to solve these expressions.
10-5 =
105 =
(-6)-2 =
124/126 =
12-3 · 126
x9/x2 =
(-2)-1 =
23/25 =

26. Problem Solving using exponents
The weight of 107 dust particles is 1 gram. How many dust particles are in 1 gram?
As of 2001, only 106 rural homes in the US had broadband internet access. How many homes had broadband internet access?
Atomic clocks measure time in microseconds. A microsecond is second. Write this number using a power of 10.

27. Exponents can be very useful for evaluating expressions, especially if you learn how to use your calculator to work with them.