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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.