Download Powers and Roots Student Notes

CHAPTER 4: POWERS AND ROOTS
Approx. 10 classes
1. Estimating Roots and Irrational Numbers– pg. 204-212 (1 class)
Assignment: pg. 206 #1-6, pg. 211-212 #1-14, 17, 20, 23
2. Mixed and Entire Radicals – pg. 213-219
Assignment: pg. 218-219 #3-18, 20-22, 24
3. Fractional Exponents and Radicals – pg.222-228 (1 class)
Assignment: pg. 227-228 #3-19, 21
4. Negative Exponents and Reciprocals- pg. 229-234
(1 class)
Assignment: pg. 233-234 #3-13, 15, 16, 18, 19
5. Applying the Exponent Laws – pg. 237-243 (1 class)
Assignment: pg. 241-243 #3-17, 19, 21, 22
6. Chapter Quiz – (1 class)
Assignment: pg. 221#1-11, pg. 236 #1-8
7. Chapter Review – pg. 246-248 (1 class)
Assignment: pg. 246-248 #1-22, 24-30, 32
8. Chapter Exam – (1 class)
1
LESSON 1: ESTIMATING ROOTS AND IRRATIONAL NUMBERS
Learning Outcome: Learn to explore decimal representations of different
roots of numbers. Learn to identify and order irrational numbers.
Work with a partner:
Since 3² = 9, 3 is a square root of 9
We write: 3 = √9
Since 3³ = 27, 3 is a cube root of 27.
3
We write: 3 = √27
Since 34 = 81, 3 is a fourth root of 81.
4
We write: 3 = √81
How would you write 5 as a square root?
A cube root? A fourth root?
The parts of a radical:
n
a is a radical
is the radical sign
a is the radicand
n is the index or order of the radical
To evaluate cube roots using a graphing calculator:
MATH: 4: 3√(
2
To evaluate a root using any index greater than 3, we need to use a twostep process.
1. Before you choose any function on the calculator, you need to first type
4
in the index you need. (if evaluating √16, you would type 4 on the screen
first)
2. MATH: 5: 𝑥√(, then type in the value you are evaluating.
Investigation:
With a partner, fill in the chart for each radical and determine if the value is
exact or approximate.
An exact value is a rational number, can be expressed as a whole number,
terminating decimal, or as a repeating decimal.
Approximate values need to be rounded, meaning the decimal goes on
infinitely without terminating or repeating.
Radical
√16
√27
16
√
81
Value
4
5.1962
0. 4̅
Is the Value Exact or Approximate?
Exact
Approximate
Exact
√0.64
3
√16
3
√27
16
√
81
3
3
√0.64
√−0.64
4
√16
4
√27
3
16
√
81
4
4
√0.64
3
Consider the values in each box:
These are rational numbers
These are not rational numbers
3
√100, √0.25, √8, 0.5
5
,
6
3
√0.24,
√9,
1
9
√2
4
√ , √12
3
√ , 0.8²
64
How are radical that are rational numbers different from radicals that are
not rational numbers? (you may want to evaluate each radical to notice the
difference between the two)
Which of these radicals are rational numbers? Which are not rational
numbers? How do you know?
64
3
4
√1.44, √81, √−27, √5, √5
Write 3 other radicals that are rational numbers. Why are they rational?
Write 3 other radicals that are not rational numbers. Why are they not
rational?
4
Radicals that are square roots of perfect squares, cube roots of perfect
cubes, and so on are rational numbers. Rational numbers have decimal
representations that either terminate or repeat.
Irrational Numbers: Q : is a number that cannot be expressed as a
terminating or repeating decimal. Irrational numbers are non-repeating
decimals. They cannot be expressed in the form
a
. b≠0
b
Examples of Q :   3.141592653
2  1.414213562
 7  2.645751311
When an irrational number is written as a radical, the radical is the exact
3
value of the irrational number; for example, √2 and √−50. We can use the
square root and cube root keys on a calculator to determine approximate
values of these irrational numbers.
Can you name the most famous irrational number?
Ex. Tell whether each number is rational or irrational. Explain your
reasoning.
a. √
25
3
b. √−30
9
Together, the rational numbers and irrational numbers form the set of real
numbers.
5
Just as living things can be classified using a series of nested sets, so can
sets of numbers:
REAL
RATIONAL
INTEGERS
IRRATIONAL
WHOLE
NATURAL
NUMBERS
Ex. Use a number line to order these numbers from least to greatest.
3
3
4
√2, √−2, √6, √11, √30
Assignment: pg. 206 #1-6
Assignment: pg. 211-212 #1-14, 17, 20, 23
6
LESSON 2: MIXED AND ENTIRE RADICALS
Learning Outcome: Learn to express entire radicals as mixed radical, and
vice versa.
We can name the fraction
3
12
in many different ways:
1
5
30
100
4
20
120
400
How do you show that each fraction is equivalent to
Why is
1
4
the simplest form of
3
3
?
12
?
12
Just as with fractions, equivalent expressions for any number have the
same value.
 √144 can be separated into its factors and rewritten:
√144 = √16 × √9
 Similarly √216 can be separated into its factors and rewritten:
3
3
3
3
√216 = √8 × √27
How could you rewrite √12 ?
3
How could you rewrite √54?
How about √128 ?
3
How about √128 ?
7
Multiplication Property of Radicals:
𝑛
𝑛
𝑛
√𝑎𝑏 = √𝑎 • √𝑏
Where n is a natural number, and a and b are real numbers.
Refer to the Perfect Square/Cube chart. In the chart, fill in all perfect
squares from 1 -22 and all perfect cubes from 1-10.
This chart will be instrumental in helping us simplifying radicals.
We can use this property to simplify square roots and cube roots that are
not perfect squares or perfect cubes, but have factors that are perfect
squares or perfect cubes.
We look for the factors of 24, with one of the factors being a perfect square
root.
1.
24
2. 72
3.
3
54 (need to use perfect cube chart!!)
8
3
4. √144
5. √63
3
6. √108
4
7. √32
9
Changing from mixed radicals to entire radicals:
√80 is an example of an entire radical.
4√3 is an example of a mixed radical.
Raise the coefficient to the index and move it back inside the radical sign
and multiply it by the radicand.
Ex. 4 5  42  5  80
Ex. 3 3 4  3 33  4  3 108
Ex. Write each mixed radical as an entire radical:
a. 7√3
3
b. 2 √4
5
c. 2 √3
Assignment: pg. 218-219 #3-18, 20-22, 24
10
LESSON 3: FRACTIONAL EXPONENTS AND RADICALS
Learning Outcome: Learn to relate rational exponents and radicals.
Work with a partner:
Complete each table. Use a calculator to complete the second column:
1
x
𝑥3
1
1
13
1
8
83
1
27
273
64
643
125
1253
x
𝑥2
1
12
4
42
9
92
16
162
25
252
1
1
1
1
1
1
1
1
For each table:
What do you notice about the numbers in the first column?
Compare the numbers in the first and second columns. What conclusions
can you make?
1
What do you think the exponent means ?
2
1
What do you think the exponent means?
3
1
1
What do you think 𝑎4 and 𝑎5 mean?
11
Powers with fractional exponents can be written as radicals in the form
1
𝑛
𝑥 𝑛 = √𝑥 , where n≠ 0. When n is even, x cannot be negative, since the
product of an even number of equal factors is always positive.
Rational Laws of Exponents: (very important)
1
a
x  x
a
a
b
x  x
b
a
or x   b x 
a
b
a
Ex. Express each power as an equivalent radical.
1
a. 642
3
b. 164
1
c. (8𝑥 2 )3
Ex. Evaluate each power without using a calculator:
1
a.
16 4
(81)
1
b. (−8)3
12
Ex. Express each radical as a power with rational exponent:
4
a. √43
5
b. √34
Ex. Evaluate: (rewrite as a radical and evaluate using a calculator)
3
a. 0.012
4
b. (−27)3
c. 0.751.2
2
Ex. Use the formula 𝑏 = 0.01𝑚3 to estimate the brain mass of each animal:
a. a moose with a body mass of 512 kg
b. a cat with body mass of 5kg
Assignment: pg. 227-228 #3-19, 21
13
LESSON 4: NEGATIVE EXPONENTS AND RECIPROCALS
Learning Outcome: Learn to relate negative exponents to reciprocals.
Two numbers with a product of 1 are reciprocals.
1
1
4
4
Since 4• = 1, the numbers 4 and are reciprocals.
2
3
2
3
2
3
Also, • = 1, so the numbers
𝑎𝑛𝑑
3
2
are also reciprocals.
We define powers with negative exponents so that previously developed
properties such as 𝑎𝑚 • 𝑎𝑛 = 𝑎𝑚+𝑛 and 𝑎0 = 1 still apply.
Apply these properties:
5−2 • 52 = 5−2+2 = 50 = 1
Since the product of 5−2 and 52 is 1, 5−2 and 52 are reciprocals.
So, 5−2 =
1
and
52
That is, 5−2 =
1
5−2
= 52
1
25
x  n is defined to be the reciprocal of x n , that is, x  n 
Ex. Evaluate each power:
a. 7−2
10 −3
b. ( )
3
14
1
xn
(x  0)
c. (−1.5)−3 (rewrite as a positive exponent and evaluate using a
calculator)
Evaluate each power without using a calculator:
5
a. 16−4
1
b.
25 −2
(36)
5
7
Ex. Use the formula 𝑣 = 0.155𝑠 3 𝑓 −6 to estimate the speed of a dinosaur
when s = 1.5 and f = 0.3
Assignment: pg. 233-234 #3-13, 15, 16, 18, 19
Checkpoint 1 and 2- pg. 220-221 and pg. 235-236 (1 class)
Assignment: pg. 221#1-11, pg. 236 #1-8
15
LESSON 5: APPLYING THE EXPONENT LAWS
Learning Outcome: Learn to apply the exponent laws to simplify
expressions.
Exponent Law Description
Algebraic representation
n a  nb  n a b
Product of Powers
To multiply powers with the same base,
add the exponents
n a  n b  n a b , n  0
Quotient of Powers
To divide powers with the same base,
subtract the exponents
n 
Power of a Power
a b
 nab
To determine the power of a power,
multiply the exponents
 m  n
Power of a Product
a
 ma  na
The power of a product is equal to the
product of the powers
Power of a Quotient
a
ma
m
   a ,n  0
n
n
The power of a quotient is equal to the
quotient of the powers
Negative Exponent Law
Rational Laws of Exponents
xn 
1
xa  a x
1
xn
( x  0)
b0 = 1
Zero Rule
Work on your own:
16
or x b   b x 
a
a
x b  b xa
a
𝑎6 𝑏9
−2
Given: ( 5 8 )
𝑎 𝑏
when a = -3 and b = 2.
Evaluate the expression by substituting the values in the expression.
Evaluate a second time, but before you substitute values in, simplify the
expression first.
Did you get the same answer?
Which strategy is more efficient? Why?
Exponents are not simplified until there are no brackets or negative
exponents in our final answer.
Ex. Simplify by writing as a single power.
 2 x3 y 2 
a.  2 6 
 4x y 
3
First simplify the
exponents in the original
equation. Then use the
negative exponent law to
create a positive
exponent. Then use the
power law to simplify the
expression.
b.  3a3b2 15a 2b5 
17
−3
4 2
c. [(− ) ]
5
5
−5
4 4
÷ [(− ) ]
5
1
−
94 • 9 4
d.
3
94
e. 𝑚4 𝑛−2 • 𝑚2 𝑛3
f.
6𝑥 4 𝑦 −3
14𝑥𝑦 2
3
g. (25𝑎4 𝑏2 )2
18
1
50𝑥 2 𝑦 4 2
h. ( 4 7 )
2𝑥 𝑦
40𝑥 4 𝑦 7
i. (
4𝑥𝑦 2
−2
)
j. (−2𝑚2 𝑛)(4𝑚𝑛−1 )
k. −8𝑥 −2 𝑦 4 𝑧 −7
19
Ex. A cone with height and radius equal has volume 18cm³. What are the
radius and height of the cone to the nearest tenth of a centimetre?
𝑉=
1
𝝅𝒓²𝒉
3
Assignment: pg. 241-243 #3-17, 19, 21, 22
Chapter Quiz – (1 class)
Chapter Review – pg. 246-248 (1 class)
Assignment: pg. 246-248 #1-22, 24-30, 32
Chapter Exam – (1 class)
20