Exponents. 1. Relate and apply the concept of exponents (incl. zero). 2. Perform calculations following proper order of operations. 3. Applying laws of.

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

.

,

.

,

TIONMULTIPLICA

repeatedundergoesthatVariableornumberthenotationlExponentiaInBase

factoraasusedis

basethetimesofnumberthenotationlExponentiaInExponent

42BASE

EXPONENTmeans 162222

IMPORTANT EXAMPLES81)3333(34 means

81)3()3()3()3()3( 4 means

27)333(33 means

27)3()3()3()3( 3 means

Variable Expressions

))()((

))()()()(()(

3

5

yyymeansy

xxxxxtionmultiplicaforsparentheseusemeansx

Substitution and EvaluatingSTEPS

1. Write out the original problem.2. Show the substitution with parentheses.3. Work out the problem.

3;4: xxifSolveExample 3;4 xxifSolve

3)4( = 64

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

22 )()( yx

22 )()( yx

22 )2()1(

541

Step 1

Step 2

Step 3

2)( yx

2)( yx

2)2()1(

9)3( 2

Step 1

Step 2

Step 3

ywx

ywx

2)1)(3(

Step 1

Step 2

Step 3

3)1)(3(

MULTIPLICATION PROPERTIESPRODUCT OF POWERS

This property is used to combine 2 or more exponential expressions with the SAME base.

53 22 )222( )22222( 82 256

))(( 43 xx ))()(( xxx ))()()(( xxxx 7x

MULTIPLICATION PROPERTIESPOWER TO A POWER

This property is used to write and exponential expression as a single power of the base.

32 )5( )5)(5)(5( 222 65

42 )(x ))()()(( 2222 xxxx 8x

MULTIPLICATION PROPERTIESPOWER OF PRODUCT

This property combines the first 2 multiplication properties to simplify exponential expressions.

2)56( )5()6( 22 9002536

3)5( xy ))()(5( 333 yx33125 yx

532 )4( xx 5323 ))(4( xx 5222 ))()(()64( xxxx

56 ))(64( xx 1164x

MULTIPLICATION PROPERTIESSUMMARY

PRODUCT OF POWERSbaba xxx

POWER TO A POWER baba xx

POWER OF PRODUCTaaa yxxy )(

ADD THE EXPONENTS

MULTIPLY THE EXPONENTS

ZERO AND NEGATIVE EXPONENTSANYTHING TO THE ZERO POWER IS 1.

27

1

3

13

9

1

3

13

3

1

3

13

13

33

93

273

33

22

11

0

1

2

3

22222

222

4

1

2

1

)2(

1)2(

2122

xxxx

xxx

8131

31

3

11

311

3

1 44

4

4

4

DIVISION PROPERTIES QUOTIENT OF POWERS

This property is used when dividing two or more exponential expressions with the same base.

))()((

))()()()((3

5

xxx

xxxxx

x

x 2

1

))((x

xx

7434

34

3

4

3 11111

xxxx

xxx

x

x

DIVISION PROPERTIESPOWER OF A QUOTIENT

12

8

43

424

3

2

)(

)(

y

x

y

x

y

x

Hard Example

3

43

2

3

2

yx

xy

343

32

)3(

)2(

yx

xy

1293

633

3

2

yx

yx

69

123

27

8

yx

yx

69

123

27

8

yx

yx6

6

27

8

x

y

ZERO, NEGATIVE, AND DIVISION PROPERTIES

Zero power 1)( 0 x

Negative Exponents

aa

aa

xx

andx

x

1

1

Quotient of powers

bab

a

xx

x

Power of a quotient

a

aa

y

x

y

x

0²=0 6²=36 12²=1441²=1 7²=49 13²=1692²=4 8²=64 15²=2253²=9 9²=81 16²=2564²=16 10²=100 20²=4005²=25 11²=121 25²=625

Exponents inOrder of Operations

1) Parenthesis →2) Exponents

3) Multiply & Divide4) 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


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


6r · 5r²

Answer:

30r³


6x · 2x²

Answer:

12x³


6x² · 6x³y⁴

Answer:

36x⁵y⁴


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


(-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 = 8/125

25

25

25

Exponent Rule:(1/a)² = 1/a²

Example: (1/7)² = 1/7² = 1/49

Exponent Rule:

a ÷aⁿ = am - 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



5⁴5

Answer:

5³(125)


2²2³

Answer:

1/2


3r³2r

Answer:

3r²2


3xy 5x²( )

2

Answer:

9y²25x²


18x⁸y⁸ 10x³

Answer:

9x⁵y⁸5


(a²)³

Answer:

a⁶


(3a²)³

Answer:

27a⁶


(2³)³

Answer:

2⁹


(8)³

Answer:

2⁹


(x⁴y⁴)³

Answer:

x¹²y¹²


(2x⁴y⁴)³

Answer:

8x¹²y¹²


(4x⁴ x⁴)³∙

Answer:

64x²⁴


(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

Competition Problems

Points: 1 minute: 5 points

1 ½ minute: 3 points2 minute: 1 point

3, 2, 1, … go!

Simplify:

(4x4y)3 (2xy3)

Answer:

128x13y6

If A = (7 – 11 + 8)131 andB = (–7 + 11 – 8)131

then what is the value of:

(7 – 13)(A+B)

Answer:

1

Simplify:

2

2

3

4

2

jh

hj

Answer:

2

4

4h

j

Evaluate forx = –2, y = 3 and z = –4:

23

432

15

5

zx

yx

Answer:

-540

If A B = (3A–B)♣3, then

what is (2 8) 6?♣ ♣

Answer:

-27,000

If a*b is defined as(ab)2 + 2b, and x y is

defined as xy2 - 2y, find 2*(3 4).

Answer:

6480

Simplify:

24 – 4(12 – 32 – 60)

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

Express in simplest form:

Answer:

Simplify:

Answer:

32

Simplify. Write the answer with negative exponents.

(abc)-3c2ba-4bc2a

Answer:

b-3c-3

Simplify.

2 2 3 2 4 2 5 2 59 2 3 4 5 6 … 60 · · · · ·

Answer:

1/900

Solve for n:

Answer:

n = 2/3

Solve for q:

.

Answer:

no solution

.

Simplify:

Answer:

Exponents. 1. Relate and apply the concept of exponents (incl. zero). 2. Perform calculations following proper order of operations. 3. Applying laws of.

Documents

n answer

positive exponents

power power of product

laws of exponents

concept of exponents

single power

negative number

odd number power