1. Positive, Negative, and Square Roots
Exponents
Positive, Negative,
and Square Roots

2. Vocabulary

3. Power Definition: A number that can be written using an exponent
Example:
The power 7² is read as seven to the second power or seven squared

4. Factor
Definition:
A number that divides into a whole number with a remainder of zero.
Example:
The factors of 12 are 1, 2, 3, 4, 6, and 12.

5. Base Definition: In a power, the number used as a factor. Example:
In 6³, the base is 6. That is, 6³ = 6 ∙ 6 ∙ 6.

6. Exponent
Definition:
In a power, the number of times the base is used as a factor.
Example:
In 5³, the exponent is 3. That is, 5³ = 5 ∙ 5 ∙ 5.

7. Squared Definition: A number raised to the second power. Example:
Five squared is written as 5², which means 5 ∙ 5.

8. Cubed Definition: A number raised to the third power Example:
Nine cubed is written as 9³, which means 9 ∙ 9 ∙ 9.

9. Examples Write each of the following statements using exponents:
Three to the fourth power
Twelve squared
Nine cubed

10. Examples Write each of the following using exponents: 3 ∙ 3 ∙3 ∙ 3
2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2
x ∙ x ∙ x ∙ x ∙ x ∙ x

11. Examples
Write each of the following in expanded form:

12. Examples
Evaluate each of the following:

13. Zero Note: Anything to the zero power is equal to 1. Examples: 5º = 1
15º = 1
2,342º = 1
xº = 1
yº = 1

14. One Note: Any number raised to the first power is that number.
Examples:
2¹ = 2
56¹ = 56
x¹ = x

15. Some Quick Review
Multiplicative Inverse: a number times its multiplicative inverse is 1 (the identity).
Another name for multiplicative inverse is reciprocal.
Examples:
The multiplicative inverse of is .

16. Negative Exponents Example: Explanation:
Since exponents are another way to write multiplication and the negative is in the exponent, to write it as a positive exponent we do the multiplicative inverse which is to take the reciprocal of the base.

17. Negative Exponents/Reciprocals

18. Examples
Rewrite each of the following in standard form.

19. Multiplying Powers Product of Powers: In words:
You can multiply powers that have the same base by adding the exponents.
Product of Powers:
In symbols:
For any number a and positive integers m and n,

20. Examples
Note: This only works if the bases are the same.

21. Practice
(3a²)(4a³)
d ∙ d³ ∙ d
m³(m³n)
(x²y²)(x³y³)

22. Dividing Powers Quotient of Powers: In words:
You can divide powers that have the same base by subtracting the exponents.
Quotient of Powers:
In symbols:
For any whole numbers m and n, and nonzero number a,

23. Examples
Note: This only works for powers that have the same base.

24. Practice 8³ 8² 9² x³ x
x³y
x²y
m²n³
mn³
45x²
15x

25. Some basics that make exponents easier to remember
Know your multiplication facts:
You should have the multiplication table from 1 x 1 to 12 x 12 committed to memory.
That means you should have them memorized and not have to use your fingers to figure them out.
If you need to work on your multiplication facts, you can make flashcards to help you practice.
You can even work on filling out tables as practice.

26. Squares You need to memorize the products of 1 through 15 squared.
1² = 1 2² = 4 3² = 9 4² = 16 5² = 25 6² = 36 7² = 49 8² = 64
9² = 81
10² = 100
11² = 121
12² = 144
13² = 169
14² = 196
15² = 225

27. Cubes You need to memorize the products of 1 through 10 cubed.
1³ = 1 2³ = 8 3³ = 27 4³ = 64 5³ = 125
6³ = 216
7³ = 343
8³ = 512
9³ = 729
10³ = 1.000

28. Explore (-5)³ (2x)² (-3xy)³ (7abc)4 (2x²)³ (5x³y²)³ (2x³y³z)6
Simplify each of the following expressions: (Hint: write the powers out.)
(-5)³
(2x)²
(-3xy)³
(7abc)4
(2x²)³
(5x³y²)³
(2x³y³z)6

29. Raising a Power to a Power
Property:
When raising a power to a power multiply the exponents.
Explanation:
When an expression that contains and exponent is written in parentheses and the parentheses have an exponent, multiply the exponents to find the power of the simplified expression

30. Examples (4v²)² = (4v²)(4v²) = 16v4
(2x³y³)³ = (2x³y³)(2x³y³)(2x³y³) = 8x9y9
(-6xy4)5 = (-6xy4)(-6xy4)(-6xy4)(-6xy4)(-6xy4) = -7776x5y20
(-2a4b5c7)8 = (-2a4b5c7) (-2a4b5c7) (-2a4b5c7) (-2a4b5c7)
(-2a4b5c7) (-2a4b5c7) (-2a4b5c7) (-2a4b5c7)
= -256a32b40c56
Do not copy this sentence, but there is a mistake in one of the examples, first to spot it and raise their hand to correct it gets a Hershey kiss :-)

31. Square Root
Remember: To square a number means to multiply the numbers by itself.
When you find the square root of a number you are looking for the factor that when multiplied by itself gave you the number.

32. Square Root (cont.) Definition: One of two equal factors of a number
This symbol refers to the square root of a number and is called a radical sign.
Example:

33. Perfect Squares Not all numbers have a whole number as a square root.
Numbers that have a whole number as a square root are referred to as perfect squares.

34. Perfect Squares (cont.)
The first fifteen perfect squares are:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, and 225.