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
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 = 8/125
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
Are you ready?3, 2, 1…lets go!
Simplify. Your answer should contain only positive exponents.
5⁴5
Answer:
5³(125)
Simplify. Your answer should contain only positive exponents.
2²2³
Answer:
1/2
Simplify. Your answer should contain only positive exponents.
3r³2r
Answer:
3r²2
Simplify. Your answer should contain only positive exponents.
3xy 5x²( )
2
Answer:
9y²25x²
Simplify. Your answer should contain only positive exponents.
18x⁸y⁸ 10x³
Answer:
9x⁵y⁸5
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
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: