CLASS NOTES 7.1 and 7.2 (part 1) nth Roots and
Rational Exponents
a
m
n
=
( a)
m
n
Evaluate the following expressions using the above rule
3
2
−2
92
32 5
64 3
Rewrite the following problems using rational exponential notation. Then
evaluate the expression using a calculator. Round to 2 decimal places.
(3 15 ) 7
( 7 −42 )2
Rewrite the following using radical notation
43
4
1
5
4
−10 3
(−10) 3
Simplify the following
2
5
7
2
3
2 3
250 1
2
2 ⋅4
1
6
Rules for Exponents
1.) Multiplying Exponents
a m ⋅ a n = a m+n
2.) Exponents Raised to an Exponent EX:
(a m ) = a mn n
3.) Product Raised to an Exponent -­‐-­‐ (ab )
m
4.) Negative Exponents (a )
−m
5.) Quotient Property =
1
am
am
= a m − n an
3
2
EX: (5 4 ) =______________________ = a ⋅ b m
21 2 ⋅ 23 2 = __________
m
EX: (9 x
EX: 36
EX: 4
1
2
) =____________________ −1 2
= __________ 45 2
= ________ 44 2
m
13
am
⎛a⎞
⎛ 8 ⎞
6.) Quotient Raised to a Power ⎜ ⎟ = m EX: ⎜
⎟
b
⎝b⎠
⎝ 27 ⎠
Practice Problems: 1
2
1
4
1.) 5 • 5 2
⎛ 1 ⎞
7
2.) ⎜ 8 2 ⋅ 2 ⎟ 3.) 1 ⎝
⎠
73
⎛ 13
⎜ 12
5.) ⎜ 1
⎜ 43
⎝
4.) 30
5
1
5
1
5
= __________ 2
⎞
⎟
⎟ ⎟
⎠
6.) 7 3 16 ⋅ 2 3 4 1
5
2
1
4
7.) 3 ⋅ 27 8.) 4
10.) 3 54 1
64 7 ⋅ 64 7
6
7
9.) 3
3
32
4
11.) 5
3
2
3
7
4⋅32
12.) 13.) 6 2
3
2 ⋅ 2 Adding and Subtracting Roots and Radicals-­‐ Add or Subtract the Following ⎛ 1⎞ ⎛ 1⎞
1.) 7⎜⎜ 6 2 ⎟⎟ + 2⎜⎜ 6 2 ⎟⎟ = ⎝ ⎠ ⎝ ⎠
4
2.) 3 16 − 3 2 = 4
4
3.) 3 64 − 3 324 7.1 AND 7.2 PART 1: HOMEWORK Evaluate the following expressions without using a calculator. 4
−
3
5
−
2
1.) 8 3 2.) 81 2 3.) 125 3 4.) (−125) 3 Evaluate the following expressions with the use of a calculator. Round to 2 decimal places. 5.)
( 8 ) 2
5
8.) 26
−
3
4
6.) 9.) ( 5)
Simplify the following expressions: 10.) 16 ⋅
3
3
( 8 ) 3
5
2
7.) (
7
)
3
−280 −4
16
4 11.) 4 81
12.) 3
5
3
⋅3
1
3
1
2 2
⎛
⎞
13.) ⎜ 125 3 ⎟ ⎝
⎠
14.) 36
−
1
2
1
15.) 4 5 ⋅ 8 5 2
1
2
16.) 3
1
1
⎛ 14 13 ⎞
17.) ⎜ 2 ⋅ 2 ⎟ ⎝
⎠
−
1
2
⎛ 52 ⎞
18. ) ⎜ 2 ⎟
⎝8 ⎠
⎛ 3 3⎞
21.) ⎜ 10 4 ⋅ 4 4 ⎟ ⎝
⎠
9⋅36
2⋅6 2
24.) 3 18 ⋅ 3 15 2
19.) 1
1
125 9 ⋅125 9
5
22.) (
3
1
4
20.) 12
6⋅4 6
)
12
25.) 3 24 ⋅ 5 2 4
3
−
3
8
−4
23.) 26.) 6 + 5 6 4
12 8
5
6
5
1
7
1
7
27.) 5(5) − 7(5) 1
1
28.) 160 2 − 10 2 29.) 2 3 81 − 3 3 7.2 Simplifying Exponents and Radicals with Variables 7
27 =
4
x4 =
Simplifying Rational Exponents and Radicals-No Denominator
Assume all variables are positive.
1.)
3
2.)
125y 6 =
1
4
3.) (625 j k ) =
8
4
3
8r 3 s 5 t 10 =
1
6 xy 2
4.)
1
3
2x z
=
−5
2
5.)
4
64d 4 e 9 f 14 =
15d 2 e 3 f
6.)
=
5df −4
7.)
4
x4
=
y8
8.)
3
x
=
y7
9.)
5
g2
=
h7
10.)
4
x 8 y 11
=
z6
Add or subtract the following Rational Exponents or Radicals.
Assume all variables are positive.
1.) 5 y + 6 y =
1
3
1
3
3.) 2 xy − 7 xy =
Homework: p. 412 #56-81 all
2.)
3
5 x 5 − x 3 40 x 2 =
4.) 33 6 y 7 + 2 y 2 3 6 y =
7.2 Practice 7.3 Power Functions and Function Operations From Larson Text, page 415 Adding and Subtracting Functions Find the sum or difference. Determine the domain of each function and the domain of the sum or difference. Create the graphs of each function and the sum or difference. Do the graphs make sense to you? f (x) = 3x1/2 g(x) = −8x1/2 1. 2. f (x) + g(x) = f (x) − g(x) = Multiplying and Dividing Functions Find the product or quotient. Determine the domain of each function and the domain of the product or quotient. Create the graphs of each function and the product or quotient. Do the graphs make sense to you? g(x) = 5x f (x) = 3x1/2 g(x)
= 1. 2. f (x)⋅ g(x) = f (x)
3. 4. 3 f (x) − 2g(x) = [ f (x)]2 + g(x) = Composition of Functions f (x) = 3x −1
Given: g(x) = 2x − 1
Determine the following compositions and their domains. Try to make sense of their graphs on your graphing calculator. 1. domain: f (g(x)) = 2. domain: g( f (x)) = 3. domain: f ( f (x)) = Applications: 1. A snowboard shop advertises that they are having a 30% off sale on all of their snowboards. When you arrive at the shop, the sales person tells you that for today only they are taking an additional 20% off of the sale price. Let x represent the original cost for the snow board. a) Write an expression that represents the sale price of the snow board when it is 30% off. b) Write an expression that represents the sale price of the snow board when you take an additional 20% off of the sale price. c) What is the price of a $375 snow board with both discounts? d) The salesperson tells you that their discounts mean you are getting a snowboard at ½ off. Do you agree? 2. During a hurricane, your basement starts to take on water due to a leak around the foundation of your house. Your sump pump kicks on to pump water out of the basement. Water is flowing in at a rate of f (x) = 5 − 2 x cubic feet per hour. Water is being pumped out of the house at a rate of ⎛ 1⎞
g(x) = ⎜ ⎟ x 2 cubic feet per hour. Observe the graph below and answer the ⎝ 50 ⎠
following questions. a) During what time interval is the water level rising in the basement? b) At what time is the water coming into the basement the same as the water pumped out of the basement? c) During what time interval is the water level decreasing in the basement? d) Set up an algebraic equation that would represent the change in the amount of water in the basement at any point in time. e) Set up an algebraic equation that would represent the time when the water coming in is equal to the water going out of the basement. Can you use your graphing calculator to solve this equation? 7.4 Inverse Functions Example 1
f(x)= 2x – 4
x
-2 -1
f(x)
f-1(x) = .5x + 2
0
1
2
x
f-1(x)
-8
-6
-4
Plot f(x) and f-1(x) on the graph below:
Plot the line y = x above.
Given the equation f(x) = 2x – 4, find the equation of its inverse.
(switch x and y, then solve for y)
Verify that the above functions are inverses.
f(f-1(x) )= x
AND
f-1(f(x)) = x
-2
0
Example 2
a.) Find the inverse of the function
f (x) =
3
x−3
b.) Graph f (x) = x − 3 and its inverse on the graph below. Then graph
the line y = x to be sure that the graphs reflect about this line.
3
c.) Verify that the above functions are inverses.
f(f-1(x) )= x
AND
f-1(f(x)) = x
Example 3
Graph f(x) = x 2
x
-2 -1
0
f(x)
Graph f(x) = x2,
1
Find then graph f-1(x) =
x
f-1(x)
2
x
f(x)
-2 -1
0
x≥0
1
2
Find then graph f-1(x) =
x
f-1(x)
Homework: p. 426 #15,19,27,29,31,33-35, 42,43,44
7.5 Graphing Square Root Functions y=
x
x
y
-4
-1
0
1
4
Domain:
Range:
Plot y =
x
y
x + 2 -4
-1
0
1
4
9
Domain: Range:
9
Domain: Range:
How does this shift y = x ? Plot y = x − 3 x
Y
-4
-1
0
1
4
How does this shift y = x ? Plot y = x + 2 x
y
-4
-3
-2
-1
2
7
Domain: Range: How does this shift y = x ? Plot y = x − 3 x
y
0
1
2
3
4
How does this shift y = x ? 7
Domain:
Range:
9
Re-plot
Plot y =
x
y
y=
x
2 x -4
-1
How does this shift
Plot y =
x
y
x
y
-4
x
y
4
9
Domain: Range:
0
1
4
9
Domain:
Range:
4
9
Domain: Range:
4
9
Domain: Range:
y = x ? − x -4
-1
0
1
y = x ? − 3 x -4
-1
How does this shift
Plot y =
1
y = x ? -1
How does this shift
Plot y =
0
1
x 2
How does this shift
Plot y =
0
1
y = x ? x + 5 + 6 x
y
How does this shift
Domain: Range:
y = x ? Standard Form of a Radical Function:
Homework: p.434 #15,16,19-22,25-29
y=a x−h+k
7.5 Writing Equations of Square Root Functions 7.5 Graphing Cube Root Functions y=3 x
x
y
-8
-1
0
1
8
Domain:
Range:
Plot y = 3 x + 2 x
y
-8
-1
0
1
Domain: Range:
8
How does this shift y = 3 x ? Plot y = 3 x − 3 x
Y
-8
-1
0
1
Domain: Range:
8
How does this shift y = 3 x ? Plot y = 3 x + 2 x
y
-10
-3
-2
-1
6
Domain: Range: How does this shift y = 3 x ? Plot y = 3 x − 3 x
y
-5
2
3
4
How does this shift y = 3 x ? 11
Domain:
Range:
y=3 x
Re-plot
Plot y = 2
x
y
3
x -8
-1
How does this shift
Plot y =
1
8
Domain: Range:
y = 3 x ? 13
x 2
x
y
-8
-1
How does this shift
Plot y = − 3
x
y
-8
Plot y = − 33
x
y
-8
1
8
Domain:
Range:
y = 3 x ? -1
0
1
8
Domain: Range:
8
Domain: Range:
y = 3 x ? x -1
How does this shift
3
0
x How does this shift
Plot y =
0
0
1
y = 3 x ? x − 1 + 7 x
y
How does this shift
Domain: Range:
y = 3 x ? Standard Form of a Cube Root Function:
Homework: p. 435 #31,32,33,35,38,39,40-43 all
y = a3 x − h + k
7.6 Solving Radical Equations
Example 1: Solving a Simple Radical Equation
3
Check:
x −4=0
Example 2: Solving an Equation with Rational Exponents
Check:
3
2
2 x = 250
Example 3: Solving an Equation with One Radical
4x − 7 + 2 = 5
Check:
Example 4: Solving an Equation with Two Radicals
Check:
6 x − x −1 = 0
Example 5: Solving an Equation with Extraneous Solutions
Check:
x − 4 = 2x
Homework: p. 441-442 #18,22,25,27,30,34,37,38,40,43,45,49,51,53
(Be sure to do the check !!)