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Mathematics 20
Module 1
Lesson 2
Mathematics 20
Irrational Numbers and Square Root Radicals
51
Lesson 2
Mathematics 20
52
Lesson 2
Irrational Numbers and Square Root
Radicals
Introduction
In this lesson a new set of numbers called irrational numbers is added to the family of
natural numbers, whole numbers, integers, and rational numbers. The irrational
numbers together with the rational numbers form the real number system whose graph is
the solid number line with no gaps.
There are many varieties of irrational numbers, but the particular kind studied in this
and the next lesson is the irrational number obtained by taking the square root of a whole
number.
The classes of numbers which form the real number system belong to a yet larger system
of numbers called the complex number system.
Whole Number Math
Triangular numbers are those which can be represented by triangles as shown by the
examples.
Which two consecutive triangular numbers have a difference of 10? What is the
difference of the differences between consecutive numbers?
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Lesson 2
Square numbers are those which can be represented by a square.
Which two consecutive square numbers have a difference of 13?
What is the difference of the differences between consecutive numbers?
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Lesson 2
Objectives
After completing this lesson, you will be able to
•
define and illustrate by example the term absolute value, and simplify expressions
that contain the absolute value of a number as well as the absolute value of a
variable.
•
define and illustrate by example an irrational number.
•
solve problems which use the terms square root radical, radicand, principal square
root, and perfect square.
•
express square root radicals in simplest mixed radical form.
•
work with a variety of given formulas that contain a squared term or a square root
radical.
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Lesson 2
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Lesson 2
2.1 Absolute Value
The Number Line
The point on the number line representing the number zero is chosen arbitrarily and this
point is called the origin. The point for the number 1 is chosen a certain distance to the
right of the origin and all other points representing integers are equally spaced, as shown
on the diagram.
The points which represent rational numbers are located so that the number represents
the distance between the point and the origin.
For example, if the scale is such that the point 1 is 1 cm from the origin then the point for
the rational number 2.5 is 2.5 cm from the origin. Very often units such as cm are not
stated. In such a case the point 1 is said to be one unit away from the origin, 2 is two
units away from the origin, etc.
There are exactly two points that are a non-zero distance from the origin. For example the
points -10 and 10 are both ten units from the origin. Only one point, the origin, is 0 units
away from the origin.
Absolute Value
The absolute value of a number is the distance between the origin and the point
representing the number. Distance is always non negative. Two vertical bars are used to
symbolize absolute value.
•
3 = 3

The absolute value of 3 is 3.
•
3 = 3

The absolute value of  3 is 3.
•
7

The absolute value of  7
•
0 = 0

The absolute value of 0 is 0.
1
1
= 7
2
2
1
1
is 7
2
2
What two numbers have absolute value equal to 7? This is the same as asking for the
solution to |x| = 7 .
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Lesson 2
The two values for x are 7 and  7 since each point representing these numbers is 7 units
away from the origin.
Symbols can be used for number systems.
•
•
•
•
•
I
N
W
Q

-
Integers ...,  2,  1, 0, 1, 2, ...
Natural numbers 1, 2, 3, 4, ...
Whole numbers 0, 1, 2, 3, 4 , ...
Rational numbers
Real Numbers
Example 1
Draw the graph of all the integers on the number line whose absolute values
are greater than or equal to 2.
Solution:
Find all the integers which are a distance greater than or equal to 2 from the origin.
This is the same as solving the inequality |x|
•
 2,
x I.
All the points to the left of  2 , including  2 , together with all the points to the
right of 2, including 2, have absolute value greater than or equal to 2.
The dark dots and the dark arrows show the graph of x  2 .
Example 2
Draw the graph on the number line of all the integers whose absolute value is
less than 3.
Solution:
This is the same as solving the inequality |x| < 3 ,
•
x  I.
All the points between  3 and 3, not including  3 and 3, have absolute values less
than 3.
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Lesson 2
Example 3
Draw the graph of all the rational numbers whose absolute value is less
than 3.
Solution:
This is the same as solving |x| < 3 ,
•
•
x  Q.
The graph is a solid line between 3 and  3 . Dots cannot be used here since rational
numbers are close together.
The empty circles or open dots on 3 and  3 show that 3 and  3 are not included in
the solution set.
Absolute value can be used to find the distance between two points on the number line.
If a and b are two points on a number line, the distance between these two points is
|a  b| or |b  a| .
•
•
The rule is to subtract the numbers in any order and take the absolute value.
The absolute value ensures that the distance is a positive number.
Example 4
Find the distance between 8 and  11 .
Solution:
8  (11) = 8  11 = 19 = 19
Take the absolute value of the difference.
or
(11)  8 =  19 = 19
Both ways result in the same answer.
The usual rules of order of operations are applied when working with absolute values.
Just as for brackets, the expression inside the absolute value sign is to be evaluated first.
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Lesson 2
Example 5
Evaluate  5  3(4 )   8 + 6  2 .
Solution:
 5  3(4 )   8 + 6  2
Write the original expression.
Evaluate inside the signs first.
=
 17   2  2
Take the absolute values.
Simplify.
=
=
17  2  2
32
Example 6
Solve for y in |x| + |y|
= 8,
if x =  3 .
Solution:
y = 8  x
Isolate |y| .
Substitute x =  3 .
Evaluate the right side.
y = 8  3
y = 8  3
y = 5
y = 5 and y =  5
There are two points on the number line which are 5 units away from the origin.
Exercise 2.1
1.
Simplify each absolute value expression.
a.
23
b.
 204
c.
9  2
d.
 16 + 8
e.
4 + 4
f.
4 + 3
g.
 20   3  22
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Lesson 2
2.
Find the distance between the given points using |a - b| .
a.
b.
c.
d.
3.
4.
5.
19 and 25
36 and  17
99 and  99
 376 and  540
Draw the graph of the solution to each equation or inequality on the number line.
Assume only integer solutions.
a.
|x| = 0
b.
|x|  2
c.
|x|  3 > 2
d.
|x| < 5
e.
|x| + 2  6
Evaluate each expression.
a.
9 4
b.
9 8 2
c.
2 1  4 + 3  7  2 7  3
d.
 9 
e.
8 2  6  3
3 4 
3
Solve for the variable in each case.
a.
|x| = 5
b.
|x|  3 =  1
c.
2x = 4
d.
2x + 3x  4x = 7
Mathematics 20
First collect similar terms.
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Lesson 2
2.2 Irrational Numbers and Square Root Radicals
Rational Numbers
The natural numbers, whole numbers, and integers are each part of the larger set of
numbers called the rational numbers.
Every rational number can be written in the fraction form
a
, where a and b
b
are integers, and b is not equal to zero.
a

Q =  a ,b  I ; b  0 
b

“such that”
Every rational number can also be written as either a repeating decimal or a terminating
decimal (the zeros repeat).
•
•
2
= 0 .181818 ...
11
1
= 0 .25000 ...
4
= 0.25

The digits 1 and 8 repeat.

Only the zeros repeat. This is a terminating
decimal.
Irrational Numbers
There are also decimal numbers that do not repeat or terminate.
An example of this is the number 2.0100100010000100000....
This number has a non repeating decimal since the number of zero digits increases by one
after each 1 digit.
A number like this cannot be written in the form of a fraction
a
and is, therefore, not a
b
rational number.
Any number whose decimal is non repeating and non terminating
is called an irrational number.
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Lesson 2
The rational numbers and the irrational numbers together form the set of real numbers
and the letter  is used to represent this set.
The number line is the graph of the set of real numbers. Each real number corresponds to
some point on the line and, conversely, each point on the line corresponds to a real
number. From now on, the number line will be called the real number line.
•
The graphs of sets of real numbers appear as heavily shaded lines, with solid (or
darkened) heads when indicating continuation in the direction of an inequality.
•
The graphs of sets of integers are shaded dots spaced one unit apart.
Example 1
Draw the graphs of the following sets.
a)
x| x > 2, x  
b)
{x | x > 2 , x  I }
Solution:
a)
x| x >
•
2, x  
The empty circle indicates that 2 is not included in the set but the graph
includes every real number larger than 2.
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Lesson 2
b)
{x | x > 2, x I}
•
The solid dots start at 3 because 2 is not included and there are no integers
between 2 and 3. The solid arrow head indicates an indefinite continuation of
the solution set, that is, all integers greater than 2.
Square Root Radicals
You know how to find the square of a number. The square of 2 is 2  2 = 4 . The square
of  2 is  2    2  = 4 . Both integers squared have a value of 4.
The reverse procedure is to find the square root of a number.
•
•
•
One square root of 4 is 2.
The other square root is  2 .
Similarly, one square root of 9 is 3 and another is  3 .
In general:
a is the square root of b if a  a = b , or a 2 = b .
Some numbers like 2, 3, and 5 do not have integer square roots but have irrational roots.
It is often not necessary to know the decimal form of the square root so the symbol a is
used to denote the positive square root of the number a.
The sign
is called the radical sign, and the number or expression under the radical
sign is called the radicand.
a is called a radical or the square root of a.
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Lesson 2
For example, the expression
10 is a radical which means the positive square root of 10.
Different calculators evaluate square roots
  differently.
If it does
not follow the procedure that is used in this course, check your manual.
Example 1
Use the square root function
on your calculator to find
2.
Solution:
clear
2 enter
display: 1.4142136...
Therefore,
2 = 1.4142136 ....
Because of space limitations, some calculators report only the first 7
decimals. This gives only an approximation to the infinite decimal. The entire
decimal, however, is non repeating and the number is irrational.
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Lesson 2
Activity 2.2
The 5 = 2.236068 . Square each decimal approximation using one more decimal place
each time.
Value
2.2 2
=
2.23 2
=
2.236 2
=
2.23606 2
2.236068 2
=
=
Notice that the squares approach the value 5 as more decimal places are included.
Principal Square Root
The number 9 has two square roots. The two square roots are 3 and  3 because:
3  3= 9
•
3  3 = 9
•
•
3 is the positive square root.
 3 is the negative square root.
•
This is written in radical form as:
3= 9
3 =  9


positive square root
negative square root
There is only one square root of zero. If a  a = 0 , a must equal 0.
0= 0
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Lesson 2
The number  9 has no real number square roots. It is impossible to find two equal
numbers that when multiplied together have a negative product. Negative numbers have
no square root.
•  9 is undefined.
Examples:
•
•
•
•
The principal square root of 49 is 49 = 7 .
The negative square root of 64 is  64 =  8 .
All the square roots of 16 are 16 ,  16 , or 4,  4 .
All the square roots of 15.21 are 15 .21 ,  15 .21 , or 3.9,  3.9 .
The above examples illustrate the following general properties of square root radicals.
Properties of Square Root Radicals
•
The radicand can be only zero or positive.
•
Each positive number c has two square roots,
•
The positive root is called the principal root and usually has no sign
attached to it.
c,  c .
Notice that the calculator gives only the principal root.
Sometimes an equation has a squared term in it.
2
x = 25
In this case, x can be equal to 25 or  25 .
Examples:
•
•
The solution to x 2 = 100 is x = 100 , x =  100 , or x =  10 .
The solution to x 2 = 42 .25 is x =  42.25 , or x =  6.5 .
Questions with square root equations are more common in real world situations. These
types of questions will be explored more in section 2.4.
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Lesson 2
The solution to the equation x 2 = c , for c > 0 , is x = c and x =  c .
The following examples will show you some of the different terminology that is used when
defining radicals. It is important to read the question to determine the root that is being
used.
•
The number which has 15 and  15 as square roots is 15  or  15  . The number
is 225.
•
The number which has
2
6 as a principal root is

•
The number which has  x as a root is  x
•
•
The solution to x = 5 is x = 5 2 = 25 .
2
If x = 2 .6 , then x = 2 .6  = 6 .76 .

2
 6
2
2
= 6.
=x .
Exercise 2.2
1.
Which numbers are rational and which are irrational?
a.
b.
c.
d.
e.
2.
3.
0.24567
49
3
1.1511511151 1115 ...
 2 .153525000 ...
Draw the graph of each set on the real number line.
a.
b.
c.
d.
 x < 2 , x  
{x | -1  x < 2, x  I}
x| x < 3.5 , x  
{x | x < 3.5, x  I}
a.
b.
c.
d.
Evaluate 169 ,  196 ,  225 .
Find the square roots of 37.4544.
Find the principal root of 625.
Find the solutions to y 2 = 28 .09 .
e.
x|  1
Find the solutions to x 2  = 16 .
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Lesson 2
4.
a.
b.
c.
d.
e.
f.
g.
What number has 7 as a root?
What number has 16 as a root?
If x = 5 , then what is x?
Find the solution to x = 3.75 .
Find the solution to  x =  10 .
2
Evaluate 3 .
 
Evaluate  36   4  .
2
2
2.3 Expressing Radicals in Simplest Form
Numbers such as 1, 4, 9, 16, can be written as a product of two equal whole numbers.
1 = 12 , 4 = 2 2 , 9 = 3 2 , 16 = 4 2
Any positive integer which can be written as a product of two equal integers is called a
perfect square.
Activity 2.3
A partial list of perfect squares is given. Continue the sequence by supplying at
least 5 more perfect squares.
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, . . .
Perfect squares will be used in simplifying radicals to make the radicand as small as
possible. Perfect squares are useful because their square roots are integers.
•
4  22  2
•
9  32  3
In general:
2
x = x, for x > 0 , x   .
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Lesson 2
Rules for multiplication and division of radicals are required for simplifying. These are
similar to rules of exponents.
a  b = ab
a
a
for a, b  0 ,  b  0 .
=
b
b
A radical is in simplest form if the radicand is made to be the smallest whole number
possible. This is done by removing perfect square factors from the radicand. The
examples illustrate how this simplification is done.
The object is to factor the radicand into as many perfect squares as possible.
Example 1
Simplify
8.
Solution:
Write the original expression.
Factor the radicand.
Use the product rule.
Simplify the radical 4 .
=
=
=
=
8
4  2
4  2
2  2
2 2
Example 2
Simplify
2450 .
Solution:
The number 2450 can be factored many different ways. One way is
2450 = 50  49 = 2  25  49 . Two of the factors, 25 and 49, are perfect squares and
can therefore be simplified. The factor 2 cannot be factored further.
Write the original expression.
Factor the radicand.
Use the product rule.
Simplify the radicals.
=
=
=
=
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70
2450
49  25  2
49  25  2
7  5  2
35 2
Lesson 2
Example 3
Simplify
900
.
2
Solution:
900
Write the original expression.
Use the quotient rule.
=
Simplify.
Factor the radicand.
Use the product rule.
Simplify the radical 225 .
=
=
=
=
=
2
900
2
450
225  2
225  2
15  2
15 2
Example 4
Simplify
3  3  3  7  7  13  13  2 .
Solution:
By grouping pairs of equal factors, perfect squares are formed.
Write the original expression.
3  3  3  7  7  13  13  2
2
2
2
3  3  7  13  2
Group equal factors.
=
Use the product rule.
=
Simplify each radical.
Regroup whole numbers and radicals.
Multiply whole numbers together.
Multiply radicals together.
= 273 6
Mathematics 20
2
2
2
3  3  7  13 
= 3  3  7  13  2
= 3  7  13  3  2
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2
Lesson 2
An entire radical consists only of the radical sign and the radicand.
Some examples of entire radicals are:
•
2

12

300
A mixed radical is the product of an entire radical with a real number not in radical form
(a coefficient).
Some examples of mixed radicals are:
•
2 12

 3 300
 5 2
•
The coefficients are 2,  3 , and 5.
These examples show that simplifying entire radicals to mixed radicals with lowest
possible radicand involves factoring into perfect squares. The reverse process of changing
mixed radicals into entire radicals requires the squaring of numbers.
Any number can be changed to entire radical form by making the square of the number
into the radicand.
Example 5
Change 2 and 7 to entire radical form.
Solution:
Square each of the numbers.
2=
2
2 =
4
7=
2
7 =
49
Example 6
Change the mixed radical 2 5 into an entire radical.
Solution:
Write the original expression.
Change the whole number into radical form.
Multiply the radicals.
2 5
= 4  5
= 4  5
= 20
The coefficient of a mixed radical may be a negative number. If this is the case, the
negative sign remains outside the radical sign when you change the mixed radical into
entire radical form.
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Lesson 2
Example 7
Change  6 7 into an entire radical.
Solution:
Write the original expression.
Change 6 to radical form.
Multiply the radicals.
6 7
=  62 
7
=  36  7
=  36  7
=  252
Exercise 2.3
1.
Write each as a square or as a product of squares.
a.
b.
c.
d.
e.
f.
2.
100
324
81
400
3  3  3  3  5  5
72  8
Express each entire radical in simplest form.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
l.
m.
2
2
3  3  5
9 7  5  5  2  2  2  2
49  13  13  2
 200
 121
18
54
90
 20
28
360
432
234
Mathematics 20
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Lesson 2
3.
Write each mixed radical in simplest form.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
4.
25 25
 9 64
2 81
3 16
3 25
5 9
 6 50  2
 3 72
 8 63
2

18
3
3

80
4
9
32
2
Express each mixed radical as an entire radical.
a.
b.
c.
d.
e.
f.
g.
2 2
8 3
3 5
6 5
50 2
 10 7
4
50
5
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Lesson 2
2.4 Applications of Radicals
The Pythagorean Theorem for right triangles requires the use of radicals.
For any right triangle, the sum of the squares of the lengths of the
legs, a and b, equals the square of the length of the hypotenuse, c.
leg 2  leg 2  hyp 2
a 2  b2  c2
The principal square root of both sides of the equation gives a formula for the length of the
hypotenuse c.
2
2
2
c = a + b
2
2
2
c = a + b
c=
2
2
a + b
The right side cannot be simplified further.
Example 1
For the given triangle, express the length of the hypotenuse as the
simplest possible mixed radical.
Solution:
Use c =
2
2
a + b , where c = x, a = 4 , b = 6 .
Write the formula.
c=
2
2
a + b
Substitute the known values.
x=
( 4 )2 + ( 6 )2
Evaluate.
x = 16 + 36
x = 52
x = 4  13
x = 2 13
Since 13 cannot be factored further, this is the simplest mixed radical.
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Lesson 2
This answer is in radical (or exact) form. A decimal approximation can be found
using a calculator.
x  42  62
clear
( 4 yx 2 + 6 yx 2 ) enter
display: 7.21110255093  7.21
Example 2
The area of a square with sides of length x is given by the
formula A  x 2 .
What must the length of each side be if the area is 500 square
units?
Solution:
It is convenient to isolate the variable x first. This is done by taking the principal square
root of both sides. Note that the length and area are always positive, so only the principal
root is considered.
Write the formula.
A = x2
Take the square root of both sides.
A=
x
x= A
2
x = 500 = 100  5
x = 100  5
x  10 5 units
Substitute A = 500 .
This answer is in radical (exact) form. A decimal approximation can be found
using your calculator.
clear
500 enter
display: 22.360679775  22.36
Mathematics 20
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Lesson 2
Exercises 2.4
1.
a.
Solve for x.
x
 12
3
b.
Solve for d.
2d
t
g
c.
Solve for w.
2 gE
V 
w
2.
Express the length of the hypotenuse in exact form.
3.
Express the length of the side of the triangle in both exact
and approximate decimal form.
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Lesson 2
4.
The formula for the distance one can see to the horizon is given by d 2  13 h , where
d is the distance to the horizon in kilometres, and h is the height in metres above
ground level a person is located.
a.
b.
5.
Isolate the variable d.
Calculate d if a person is observing from the top of a 400 m building.
Isolate r in the formula for the area of a circle A = r 2 .
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Lesson 2
Conclusion
In the introduction you were given a Whole Number Math problem. The solution to
this problem is:
•
Two consecutive triangular numbers that have a difference of 10 are 45 and 55.
The difference of the differences is 1.
•
Two consecutive square numbers that have a difference of 13 are 36 and 49. The
difference of the differences is 2.
Summary
The following is a list of concepts that you have learned in this lesson:
•
The absolute value of a number can be thought of as the distance between the origin
and the point representing the number on the number line.
•
Equations and inequalities with absolute values have the following solutions:
| x| < c
| x| = c
| x| > c
•
The radicand can only be zero or a positive number.
•
Each positive number c has two square roots,
•
The positive root is called the principal root and usually has no sign attached to it.
•
The calculator gives only the principal root.
•
The solution to the equation x 2 = c , for c > 0 , is x =
•
The solution to the equation
Mathematics 20
c,  c .
c , and x =  c .
x = a, is x = a 2 .
79
Lesson 2
Answers to Exercises
Exercise 2.1
1.
a.
b.
c.
d.
e.
f.
g.
23
204
7
8
0
7
5
2.
a.
b.
c.
d.
6
53
198
164
3.
a.
b.
c.
d.
e.
4.
5.
Mathematics 20
a.
b.
c.
 36
 144
2  3   10  2 4  8
d.
e.
9  3 4 3   3 3
a.
b.
c.
d.
x
x
x
x
 27
 88
=
=
=
=
5
2
2
7
80
Lesson 2
Exercise 2.2
1.
2.
a.
rational
b.
rational
c.
irrational
d.
irrational
e.
rational
a.
b.
101
c.
d.
Exercise 2.3
Mathematics 20
3.
a.
b.
c.
d.
e.
13, ±14,  15
±6.12
25
±5.3
±2
4.
a.
b.
c.
d.
e.
f.
g.
7
256
25
14.0625
100
3
144
1.
a.
b.
c.
d.
e.
f.
10
2
18
2
9
20 2
2 2 2
3 3 5
2 2 2
2 3 4
2
81
Lesson 2
2.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
l.
m.
15 3
60 7
91 2
 10 2
 11
3 2
3 6
3 10
2 5
2 7
6 10
12 3
3 26
3.
a.
b.
d.
e.
f.
g.
25 5   125
 72
29 3
 
34 2
1
 60
 3 36  2  18 2
 8 9  7  8 3  7  24 7
h.
i.
j.
2 2
3 5
18 2
a.
b.
c.
d.
e.
f.
4 2  8
64  3  192
 45
180
5000
 100  7   700
16
 50  32
25
c.
4.
g.
Mathematics 20
82
Lesson 2
Exercise 2.4 1.
x
 12
3
a.
x
 144
3
x  432
2d
g
2d
t2 
g
2
t g  2d
t
b.
d
t2g
2
2 gE
w
2
gE
V2 
w
2
V w  2 gE
2 gE
w
V2
V 
c.
2.
x  6 2  14 2
x  232
x  2 58
3.
x  15 2  3 2
x  225  9
x  216
 14 .7
x 6 6
4.
a.
b.
d  13 h
d  13 400 
d  20 13
5.
Mathematics 20
r2 
A

r
A

83
Lesson 2
Mathematics 20
84
Lesson 2
Mathematics 20
Module 1
Assignment 2
Mathematics 20
85
Lesson 2
Mathematics 20
86
Lesson 2
Optional insert: Assignment #2 frontal sheet here.
Mathematics 20
87
Lesson 2
Mathematics 20
88
Lesson 2
Assignment 2
Values
(40)
A.
Multiple Choice: Select the best answer for each of the following and place a
() beside it.
1.
The absolute value of 16 is ***.
____
____
____
____
2.
5.
only 2
only 4
 2
±2
a.
b.
c.
d.
0
10
 10
25
If 2 x  3 y = 6 , then x is equal to ***.
____
a.
6+ 3y
____
b.
____
c.
____
d.
6 + 3y
3
6+
y
2
6+ 3y
2
4 5 expressed as an entire radical is ***.
____
____
____
____
Mathematics 20
a.
b.
c.
d.
The value of  5   5 is ***.
____
____
____
____
4.
16
±4
±16
4
If the absolute value of a number is 2, the number is ***.
____
____
____
____
3.
a.
b.
c.
d.
a.
b.
c.
d.
20
9
80
10
89
Lesson 2
6.
12 2 expressed as an entire radical is ***.
____
____
____
____
7.
10.
5 12
10 30
30 10
10 3
a.
b.
c.
d.
18 10
6 5
3 20
2 45
The graph of y > 2 , y  I is ***.
____
a.
____
b.
____
c.
____
d.
The one rational number is ***.
____
____
____
____
Mathematics 20
a.
b.
c.
d.
180 expressed as a mixed radical in simplest form is ***.
____
____
____
____
9.
24
288
48
144
300 written as a mixed radical in simplest form is ***.
____
____
____
____
8.
a.
b.
c.
d.
a.
b.
c.
d.
2
3
4
5
90
Lesson 2
11.
The graph of |x| < 3 , x   is ***.
____
a.
3 0 3
____
b.
____
c.
____
d.
3
12.
a.
b.
c.
d.
 10
5 2
50
5 2
a.
b.
c.
d.
x = 72 is ***.
5184
144
6 2
6 2
The expression 3 2  2 14 
____
____
____
____
Mathematics 20
2
71
71
71
 71
The complete solution to
____
____
____
____
15.
a.
b.
c.
d.
The complete solution to 2 x 2 = 100 is ***.
____
____
____
____
14.
3
The principal square root of 71 is ***.
____
____
____
____
13.
0
a.
b.
c.
d.
2
in simplest form is equal to ***.
7
6 8
12 2
 12 2
12
91
Lesson 2
16.
The mixed radical 8 3 in entire radical form is ***.
____
____
____
____
17.
 1352
 676
 26 2
 13 2
a.
b.
c.
d.
20
5 10
50
10 5
The diagonal of a square is
____
____
____
____
20.
a.
b.
c.
d.
The length of the diagonal of a rectangle with sides 15 and 5 is ***.
____
____
____
____
19.
192
24
72
192
The mixed radical  2 338 in simplest radical form is ***.
____
____
____
____
18.
a.
b.
c.
d.
a.
b.
c.
d.
If T = 2 
128 m
16 m
64 m
8m
l
, then l is equal to ***.
g
a.
gT2
4 2
____
b.
____
c.
____
d.
g T2
4 2
gT2
2
T g
____
128 m . Each side is ***.
2
Mathematics 20
2
92
Lesson 2
Part B can be answered in the space provided. You also have the option to
do the remaining questions in this assignment on separate lined paper. If
you choose this option, please complete the entire question on the separate
paper. Evaluation of your solution to each problem will be based on the
following.
(40)
B.
•
A correct mathematical method for solving the problem is shown.
•
The final answer is accurate and a check of the answer is shown where
asked for by the question.
•
The solution is written in a style that is clear, logical, well organized,
uses proper terms, and states a conclusion.
1.
Evaluate  3  5  27   36  + 4  4 .
2.
Evaluate x  y  x 2  y2 , if x = 3 and y =  2 .
3.
Draw the graph of
a.
|x| < 2.5 , x  
b.
|x|  1.6 , x  I
(4 each)
Mathematics 20
93
Lesson 2
4.
Find the distance between  3 and  15 using a  b .
5.
Find the solution to
2
a.
x = 6 .25 .
b.
x = 3.56 .
6.
Simplify
5 49  2 25
.
25
7.
Mathematics 20
Express
1575 as a mixed radical in simplest form.
94
Lesson 2
17
75 as an entire radical.
5
8.
Express
9.
The distance in metres a free falling object travels is given by the
1
equation d  gt 2 , where t is in seconds and g which is the
2
acceleration due to gravity is 9.8.
a.
b.
10.
Isolate the variable t.
How long does it take an object to fall 1000 m?
The period of a pendulum in seconds is given by the formula
l
, where l is the length of the pendulum in metres and g is
T = 2
g
the acceleration due to gravity.
Calculate to two decimal places the acceleration due to gravity if it
takes a 10 m long pendulum 6.35 seconds to complete one cycle; i.e.,
the period is 6.35 s.
Mathematics 20
95
Lesson 2
Answer Part C on separate lined paper. Please include any tables or graphs
that you are required to do with the assignment.
(20)
C.
1.
2.
a.
Find all twelve integer ordered pairs that satisfy the equation
|x| + |y| = 3 .
Organize the ordered pairs into a table of values.
b.
Plot the ordered pairs on the attached grid and connect the
points.
c.
What geometric figure is formed?
This example shows how to locate the irrational numbers 2 and 3
on the number line by use of a compass and a ruler.
You are asked to locate the irrational numbers 5 , 6 , and 7 by the
same method.
Example:
On the real number line draw a right triangle with sides
of length 1. By Pythagoras Theorem the hypotenuse is of
length 2 . With a compass centred at the origin and
radius the length of the hypotenuse, draw an arc
intersecting the real line. Label the point of intersection
2.
Next, draw a right triangle with base length 2 , and
height 1. By Pythagoras Theorem the hypotenuse is 3 .
With a compass centred at the origin and radius the
length of the hypotenuse draw an arc intersecting the real
line at the point labelled 3 .
100
Mathematics 20
96
Lesson 2
y
x
Mathematics 20
97
Lesson 2