Download 0 Chapter 4 Notes Package1.jnt

Foundations & Pre-Calculus 10
Chapter 4
DATE
TOPIC
ASSIGNMENT
4.1 Investigation and #4
4.2 Irrational Numbers
4.3 Mixed and Entire Radicals
4.4 Fractional Exponents and
Radicals
4.5 Negative Exponents and
Reciprocals
4.6 Applying the Exponent Laws
Review
Chapter 4 Test
1
Foundations & Pre-Calculus 10
4.1 Math Lab Estimating Roots
Make Connections
Show how you would write each as a square root, cube root, and fourth root.
Square root
Cube root
Fourth root
Show 3 as …
Show 4 as …
Show 5 as …
Estimating Radicals
Ex. #1: Estimate the following radicals to 1 decimal place.
(a)
20
(b)
3
(c)
34
Check: Using your calculator.
2
4
40
Foundations & Pre-Calculus 10
Complete the following table.
Radical
Radicand
Index
Words
√81
√16
16
81
√0.64
√16
√27
16
81
√0.64
√16
√27
16
81
√0.64
3
Value
Exact or
Approximate
Foundations & Pre-Calculus 10
4.2 Irrational Numbers
Investigation:
Use your calculator to find the decimal representation of the following.
Rational Numbers
Not Rational Numbers
√100 =
√32 √0.25 √2 √0.24 √9 Rational Numbers
• can be written as a _______________ , , where m and n are integers
• have decimal representations that either ______________________ or
_____________________.
• Radicals that are square roots of perfect _________________, cube roots of
perfect ________________ and so on
Irrational Numbers
• cannot be written as a _______________ , , where m and n are integers
• the decimal representation neither _________________________ nor
_______________________
When an irrational number is written as a radical, the radical is the
value of the irrational number.
We can use the square root and cube root keys on a calculator to determine the
values of the irrational numbers.
Ex # 1: Tell whether each number is rational or irrational.
Explain how you know.
a) b) √14
4
c) Foundations & Pre-Calculus 10
Together, the rational numbers and irrational numbers form the
__________________________________________________
Real Numbers
Ex. # 2: Use a number line to order these numbers from least to greatest.
√13, √18, √9, √27, √5
5
Foundations & Pre-Calculus 10
4.3 Mixed and Entire Radicals
Radicals like 18 , 3 24 , 2 , x are called ______________________.
Radicals like 3 2 ,23 − 8 ,5 14 , a x are called _____________________.
Ex. #1: Simplify the following.
(a) √16 ! 9
(b) √16 ! √9
Multiplication Property of Radicals
√#$ =
%
Where n is a natural number, and a and b are real numbers.
We can use this property to simplify square roots and cube roots that are not
perfect squares or perfect cubes, but have ___________________________ that
are perfect squares or perfect cubes.
To change from entire radicals to mixed radicals:
1._______________________________________________________
2._______________________________________________________
3. ______________________________________________________
A radical is in simplest form when the _____________________________ has no
_______________________________ factors.
6
Foundations & Pre-Calculus 10
For example:
Write the factors of 24 below:
One factor is a perfect square. It is
.
Therefore we can show √24 as follows:
As well, we can show √24 as follows:
But we cannot simplify √24 because 24 has no factors that can be written as a
fourth power.
Ex #2: Simplify each radical.
(a) 18
(b)
(d)
48
(e)
3
50
(c)
54
(f)
7
3
16
288
Foundations & Pre-Calculus 10
To change from mixed radicals to entire radicals we:
1.____________________________________________
2.____________________________________________
Ex #3: Write each mixed radical as an entire radical.
(a) 4 3
(c) 23 5
(b) 5 11
Ex. #4: Without using a calculator arrange the following radicals from least to
greatest.
7 3 ,9 2 ,5 6 , 103 ,3 17
8
Foundations & Pre-Calculus 10
4.4 Fractional Exponents and Radicals
Powers with Rational Exponents
When ‘m’ and ‘n’ are natural numbers, and ‘x’ is a rational number,
'
%
& =
'
%
& =
and
=
=
Another (perhaps silly) “helper” to remember where everything goes:
()*
& +,,*-
√&
Hat goes on the top of your body, and is therefore seen in the
__________________ of the exponent. Boots go on the bottom of your body
and is therefore seen in the _____________________ of the exponent.
So the base goes home, leaves the boots at the door and hangs up the hat.
Ex. #1: Evaluate each without using a calculator.
a) 27
.
b) 0.49
.
c) 0641
d)
9
.
/
2 3
.
/
Foundations & Pre-Calculus 10
Ex. #2:
/
a) Write 40
in radical form in two ways.
b) Write √3 and (√25)2 in exponent form.
Ex. #3: Evaluate each.
a) 0.043/2
b) 274/3
c) (-32)0.4
d) 1.81. 4
10
Foundations & Pre-Calculus 10
4.5 Negative Exponents and Reciprocals
Recall:
• The reciprocal of
is
• The reciprocal of 4 is
Powers with Negative Exponents
When ‘x’ is any non-zero number and ‘n’ is a rational number,
& 4 is the reciprocal of & .
That is, & 4 =
and
5 6%
=
,x≠0
Ex. #1: Evaluate each power
a) 34
b) 0.3-4
c) 0 14
11
Foundations & Pre-Calculus 10
Ex. #2: Evaluate each power without using a calculator.
a) 8
6/
4
b)2 3
/
**Never flip the _______________ only the _________.**
Ex. #3: Paleontologists use measurements from fossilized dinosaur tracks and the
:
; to
formula 7 0.1558 9
estimate the speed at which the dinosaur travelled.
In the formula, v is the speed in metres per second, s is the distance between
successive footprints of the same foot, and f is the foot length in metres. Use the
measurements in the diagram to estimate the speed of the dinosaur.
4
12
Foundations & Pre-Calculus 10
Revisiting the Exponent Laws
1. Multiplication & ⋅ & _______
# ⋅ # 0____ ! ____ ! ____ ! ____10____ ! ____1 2. Division & > & ________
$ > $ 0____ ! ____ ! ____ ! ____ ! ____ ! ____1
0____ ! ____ ! ____ ! ____1
3. Power of a Power 0& 1 ________
0? 1 0____ ! ____1
0____ ! ____10____ ! ____10____ ! ____1 4. Power of a Product 0&@1 ___________
0A91 0______10______10______1
____ ! ____ ! ____ ! ____ ! ____ ! ____ 5 5. Power of a Quotient 2 3 B
C 2D 3 23 23 23 =
6. Zero Exponent: & E ______
Note the difference between the following:
1. 041
2. 4
13
Foundations & Pre-Calculus 10
From Math 9 Chapter 2 Review:
More Practice
1. 0& @ 10& @ 1
2.
) F;
3.
05 1
4. 0# $ 1 0#$ 1
)
F
5.
051/
14
05 % B 1/
05 % B 1
Foundations & Pre-Calculus 10
4.6 Applying the Exponent Laws
Recall the exponent laws for integer bases and whole number exponents.
Product of Powers:
Power of a product:
Quotient of Powers:
Power of a quotient:
Power of a power:
) ; FG
What is the value of 2 H 3
) F
4
when # 3 and $ 2?
We can use the exponent laws to _________________________ expressions that
contain rational number bases. It is a convention to write a simplified power with
a
exponent.
15
Foundations & Pre-Calculus 10
Ex. #1: Simplify by writing as a single power. Explain the reasoning.
(Don’t write too big)
16
Foundations & Pre-Calculus 10
Ex. #2: Simplify. Explain the reasoning.
Ex. #3: Simplify. Explain the reasoning.
17
Foundations & Pre-Calculus 10
Ex. #4: A sphere has a volume of 425 m3. What is the radius of the sphere to the
nearest tenth of a metre?
18