Exponents

1. Relate and apply the concept of exponents (incl. zero). 2. Perform calculations following proper order of operations. 3. Applying laws of exponents to compute with integers. 4. Naming square roots of perfect squares through 225.

EXPONENT LAWS

Basic Terminology BASE EXPONENT means

IMPORTANT EXAMPLES

Variable Expressions

Substitution and Evaluating STEPS 1.Write out the original problem. 2.Show the substitution with parentheses. 3.Work out the problem. = 64

Evaluate the variable expression when x = 1, y = 2, and w = -3 Step 1 Step 2 Step 3 Step 1 Step 2 Step 3 Step 1 Step 2 Step 3

MULTIPLICATION PROPERTIES PRODUCT OF POWERS This property is used to combine 2 or more exponential expressions with the SAME base.

MULTIPLICATION PROPERTIES POWER TO A POWER This property is used to write and exponential expression as a single power of the base.

MULTIPLICATION PROPERTIES POWER OF PRODUCT This property combines the first 2 multiplication properties to simplify exponential expressions.

MULTIPLICATION PROPERTIES SUMMARY PRODUCT OF POWERS POWER TO A POWER POWER OF PRODUCT ADD THE EXPONENTS MULTIPLY THE EXPONENTS

ZERO AND NEGATIVE EXPONENTS ANYTHING TO THE ZERO POWER IS 1.

DIVISION PROPERTIES QUOTIENT OF POWERS This property is used when dividing two or more exponential expressions with the same base.

DIVISION PROPERTIES POWER OF A QUOTIENT Hard Example

ZERO, NEGATIVE, AND DIVISION PROPERTIES Zero power Negative Exponents Quotient of powers Power of a quotient

0²=0 6²=36 12²=144 1²=1 7²=49 13²=169 2²=4 8²=64 15²=225 3²=9 9²=81 16²=256 4²=16 10²=100 20²=400 5²=25 11²=121 25²=625

Exponents in Order of Operations 1) Parenthesis →2) Exponents 3) Multiply & Divide 4) Add & Subtract

Exponents & Order of Operations

Contest Problems Are you ready? 3, 2, 1…lets go!

180 – 5 · 2²

Answer: 160

Evaluate the expression when y= -3 (2y + 5)²

Answer: 1

-3²

Answer: -9

Warning!!! The missing parenthesis makes all the difference. The square of a negative & the negative of a square are not the same thing! Example: (-2)² ≠ -2²

Contest Problems

Are you ready? 3, 2, 1…lets go!

8 ( 6² - 3(11) ) ÷ 8 + 3

Answer: 6

Evaluate the expression when a= -2 a² + 2a - 6

Answer: -6

Evaluate the expression when x= -4 and t=2 x²(x-t)

Answer: -96

Exponent Rule: a ∙ aⁿ = a m + nm Example2: 2³ ∙ 2² = 2³⁺² = 2⁵ = 32 Example1: 2 ∙ 2 = 2¹⁺¹ = 2² = 4

Simplify (in terms of 2 to some power). Your answer should contain only positive exponents. 4² · 4²

Answer: 2⁸

Simplify (in terms of 2 to some power). Your answer should contain only positive exponents. 2 · 2² · 2²

Answer: 2⁵

Simplify. Your answer should contain only positive exponents. 2n⁴ · 5n ⁴

Answer: 10n⁸

Simplify. Your answer should contain only positive exponents. 6r · 5r²

Answer: 30r³

Simplify. Your answer should contain only positive exponents. 6x · 2x²

Answer: 12x³

Simplify. Your answer should contain only positive exponents. 6x² · 6x³y⁴

Answer: 36x⁵y⁴

Simplify. Your answer should contain only positive exponents. 10xy³ · 8x⁵y³

Answer: 80x⁶y⁶

Simplify Completely. Your answer should not contain exponents. 3⁵ · 3¯⁵

Answer: 1

(-4)³

Answer: -64

(-2)⁴

Answer: 16

Important! * If a negative number is raised to an even number power, the answer is positive. * If a negative number is raised to an odd number power, the answer is negative.

Contest Problem Are you ready? 3, 2, 1…lets go!

(-1) + 1 (5²) (2⁵)

Answer: 0

Exponent Rule: (ab)² = a²b² Example: (4·6)² = 4²·6²

Exponent Rule: (a/b)² = a²/b² Example: (7/12)² = 7²/12² = 49/144

Exponent Rule: (a÷b)ⁿ = aⁿ÷bⁿ = aⁿ/bⁿ Example: (2÷5)³ = (2÷5)·(2÷5)·(2÷5) = (―)·(―)·(―) =(2·2·2)/(5·5·5) =2³/5³ = 8/

Exponent Rule: (1/a)² = 1/a² Example: (1/7)² = 1/7² = 1/49

Exponent Rule: a ÷aⁿ = a m - nm Example: 2⁵ ÷ 2² = 2⁵¯² = 2³ = 8

Exponent Rule: (a )ⁿ = a Example: ( 2² )⁵ = 2 = 2¹⁰ = 1,024 m m · n 2·5

Exponent Rule: a⁰ = 1 Examples: ( 17 )⁰ = 1 ( 99 )⁰ = 1

Exponent Rule: (a)¯ⁿ = 1÷aⁿ Example: 2¯⁵ = 1 ÷ 2⁵ = 1/32

Problems Are you ready? 3, 2, 1…lets go!

Simplify. Your answer should contain only positive exponents. 5⁴ 5

Answer: 5³ (125)

Simplify. Your answer should contain only positive exponents. 2² 2³

Answer: 1/2

Simplify. Your answer should contain only positive exponents. 3r³ 2r

Answer: 3r² 2

Simplify. Your answer should contain only positive exponents. 3xy 5x² () 2

Answer: 9y² 25x²

Simplify. Your answer should contain only positive exponents. 18x⁸y⁸ 10x³

Answer: 9x⁵y⁸ 5

Simplify. Your answer should contain only positive exponents. (a²)³

Answer: a⁶

Simplify. Your answer should contain only positive exponents. (3a²)³

Answer: 27a⁶

Simplify. Your answer should contain only positive exponents. (2³)³

Answer: 2⁹

Simplify. Your answer should contain only positive exponents. (8)³

Answer: 2⁹

Simplify. Your answer should contain only positive exponents. (x⁴y⁴)³

Answer: x¹²y¹²

Simplify. Your answer should contain only positive exponents. (2x⁴y⁴)³

Answer: 8x¹²y¹²

Simplify. Your answer should contain only positive exponents. (4x⁴∙x⁴)³

Answer: 64x²⁴

Simplify. Your answer should contain only positive exponents. (4n⁴∙n)²

Answer: 16n¹⁰

Simplify the following problems completely. Your answer should not contain exponents. Example: 2³·2² = 2⁵ = 32

-3 - (1)¯⁵

Answer: -4

(2)¯³

Answer: 1/8

(-2)¯³

Answer: - 1/8

-2 ⁽¯⁴⁾

Answer: - 1/16

(2) ¯³ · (-16)

Answer: -2

56 · (2)¯³

Answer: 7

56 ÷ (2)¯³

Answer: 448

1 ÷ (-3)¯²

Answer: 9

( 2² )³ · (6 – 7)² - 2·3² ÷ 6

Answer: 61

-6 - (-4)(-5) - (-6)

Answer: -20

2 ( 10² + 3 · 18 ) ÷ ( 5² ÷ 2¯² )

Answer: 3.08

Simplify: (x⁴y¯²)(x¯¹y⁵)

Answer: x³y³

Solve for x: (4³)⁷ = 4 x

Answer: 21

Solve for x: 2 x = 2⁵·2⁹

Answer: 14

Solve for x: 5 x = 5⁹ 5⁴

Answer: 5

Scientific Notation

How wide is our universe? 210,000,000,000,000,000,000,000 miles (22 zeros) This number is written in decimal notation. When numbers get this large, it is easier to write them in scientific notation.

Scientific Notation A number is expressed in scientific notation when it is in the form a x 10 n where a is between 1 and 10 and n is an integer

Write the width of the universe in scientific notation. 210,000,000,000,000,000,000,000 miles Where is the decimal point now? After the last zero. Where would you put the decimal to make this number be between 1 and 10? Between the 2 and the 1

2. 10,000,000,000,000,000,000,000. How many decimal places did you move the decimal? 23 When the original number is more than 1, the exponent is positive. The answer in scientific notation is 2.1 x 10 23

Express in scientific notation. Where would the decimal go to make the number be between 1 and 10? 9.02 The decimal was moved how many places? 8 When the original number is less than 1, the exponent is negative x 10 -8

Write in scientific notation x x x x 10 5

1) Express 1.8 x in decimal notation ) Express 4.58 x 10 6 in decimal notation. 4,580,000

3) Evaluate. Write in scientific notation. 4.5 x x x 10 -3

4) Evaluate. Write in scientific notation. 7.2 x x

Write (2.8 x 10 3 )(5.1 x ) in scientific notation x x x x 10 11

Write in PROPER scientific notation. (Notice the number is not between 1 and 10) 8) x x ) x 10 4 on calculator: x 10 2

Write x 10 5 in scientific notation x x x x x x x 10 8

Rational Exponents Fraction Exponents

Radical expression and Exponents By definition of Radical Expression. The index of the Radical is 3.

How would we simplify this expression? What does the fraction exponent do to the number? The number can be written as a Radical expression, with an index of the denominator.

The Rule for Rational Exponents

Write in Radical form

Write each Radical using Rational Exponents

What about Negative exponents Negative exponents make inverses.

What if the numerator is not 1 Evaluate

What if the numerator is not 1 Evaluate

For any nonzero real number b, and integer m and n Make sure the Radical express is real, no b<0 when n is even.

Simplify

Competition Problems Points: 1 minute: 5 points 1 ½ minute: 3 points 2 minute: 1 point 3, 2, 1, … go!

Evaluate: 4 5/2

Answer: 32

Simplify: (4x 4 y) 3 (2xy 3 )

Answer: 128x 13 y 6

Evaluate: (-8) -4/3

Answer: 1/16

Solve for x: x 3 = 1 / 64

Answer: 1/4

Solve for x: 3 (-x) = 9²·3 27²

Answer: 1

If A = (7 – ) 131 and B = (– – 8) 131 then what is the value of: (7 – 13) (A+B)

Answer: 1

Simplify:

Answer:

Solve for x: 125 = 25 (- ³ ) 5 x

Answer: -9

Solve for x: 2 x+2 · 4 x-2 = 16 x

Answer: -2

Write in scientific notation:

Answer: 1.6 × 10 7

Evaluate for x = –2, y = 3 and z = –4:

Answer: -540

If A ♣ B = (3A–B) 3, then what is (2 ♣ 8) ♣ 6?

Answer: -27,000

Write in standard form: (2.436 × 10 6 ) (1.2 × 10 8 )

Answer:

If f (x) = x +1 and g(x) = (x 2 − 2) 2 find: g( f (3))

Answer: 196

If a*b is defined as (ab) 2 + 2b, and x y is defined as xy 2 - 2y, find 2*(3 4).

Answer: 6480

Simplify: 24 – 4(12 – 3 2 – 6 0 )

Answer: 16

If x = the GCF of 16, 20, and 72 and y = the LCM of 16, 20, and 72, what is xy?

Answer: 2880

What is the value in scientific notation of:

Answer:

Express in simplest form:

Answer:

Simplify:

Answer: 32

Simplify. Write the answer with negative exponents. (abc) -3 c 2 b a -4 bc 2 a

Answer: b -3 c -3

Simplify. Write the answer with negative exponents. x 2 y -2 4p 0 x -5 z 2 3x -4 y 2 p 0 z 2 p 0 y -2 z -2 p

Answer: 4/x 3 - 3/x 2

Simplify … 60 ·····

Answer: 1/900

Solve for n:

Answer: n = 2/3

Solve for q:.

Answer: no solution

. Simplify:

Answer: