ExponentsPositive, Negative, and Square Roots

Vocabulary

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

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.

Base

Definition:

• In a power, the number used as a factor.

Example:

• In 6³, the base is 6. That is, 6³ = 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.

Squared

Definition:

• A number raised to the second power.

Example:

• Five squared is written as 5², which means 5 ∙ 5.

Cubed

Definition:

• A number raised to the third power

Example:

• Nine cubed is written as 9³, which means 9 ∙ 9 ∙ 9.

Examples

Write each of the following statements using exponents:

• Three to the fourth power

• Twelve squared

• Nine cubed

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

Examples

Write each of the following in expanded form:

67

45

210

Examples

Evaluate each of the following:28

310

52

Zero

Note:

• Anything to the zero power is equal to 1.

Examples:

• 5º = 1• 15º = 1• 2,342º = 1• xº = 1• yº = 1

One

Note:

• Any number raised to the first power is that number.

Examples:

• 2¹ = 2• 56¹ = 56• x¹ = x

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 .– The multiplicative inverse of is .

43

34

97

79

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.

22 1

xx

Negative Exponents/Reciprocals

Examples

Rewrite each of the following in standard form.

28

52

39

47

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,

nmnm aaa

Examples

• Note: This only works if the bases are the same.

96363 8888888888888

53232 555555555

53232 xxxxxxxxx

Practice

1. (3a²)(4a³)

2. d ∙ d³ ∙ d

3. m³(m³n)

4. (x²y²)(x³y³)

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,

nmn

m

aaa

Examples

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

3585

8

7777777

7777777777

2242

4

5555

555555

3374

7

4

61

666666666

666666

3585

8

xxxxxxx

xxxxxxxxxx

Practice

1. 8³8²

2. 9²9²

3. x³x

4. x³yx²y

5. m²n³mn³

6. 45x²15x

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.

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

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

ExploreSimplify each of the following expressions: (Hint: write the powers

out.)

1. (-5)³

2. (2x)²

3. (-3xy)³

4. (7abc)4

5. (2x²)³

6. (5x³y²)³

7. (2x³y³z)6

Raising a Power to a PowerProperty:

• 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

Examples1. (4v²)² = (4v²)(4v²) = 16v4

2. (2x³y³)³ = (2x³y³)(2x³y³)(2x³y³) = 8x9y9

3. (-6xy4)5 = (-6xy4)(-6xy4)(-6xy4)(-6xy4)(-6xy4) = -7776x5y20

4. (-2a4b5c7)8 = (-2a4b5c7) (-2a4b5c7) (-2a4b5c7) (-2a4b5c7) (-2a4b5c7) (-2a4b5c7) (-2a4b5c7) (-2a4b5c7)

= -256a32b40c56

1. 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 :-)

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.

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:

525

416

39

24

11

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.

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.