Module 1
(Effective Alternative Secondary Education)
MATHEMATICS II
MODULE 1
Radical Expressions
BUREAU OF SECONDARY EDUCATION
Department of Education
DepEd Complex, Meralco Avenue, Pasig City
Module 1
Radical Expressions
What this module is about
This module is about radical expressions. This module will require you to recall your knowledge on exponents and how to relate it to radicals. You will start with the simple basic lessons on squares and cubes.
What you are expected to learn
This module is designed for you to demonstrate knowledge and skill in simplifying, performing operations and solving equations and problems involving radical expressions.
1. Identify expressions which are perfect squares or perfect cubes.
2. Find the square root or cube root of expressions
3. Given a number in the form [pic]where x is not a perfect nth root, name two rational numbers between which it lies.
How much do you know
A. Find the indicated square roots of the following.
1. [pic] 6. [pic]
2. [pic] 7. [pic]
3. [pic] 8. [pic]
4. [pic] 9. [pic]
5. [pic] 10. [pic]
B. Find the cube roots of the following:
1. [pic] 6. [pic]
2. [pic] 7. [pic]
3. [pic] 8. [pic]
4. [pic] 9. [pic]
5. [pic] 10. [pic]
C. Name the two rational numbers between which it lies.
1. [pic] 6. [pic]
2. [pic] 7. [pic]
3. [pic] 8. [pic]
4. [pic] 9. [pic]
5. [pic] 10. [pic]
What you will do
Radical Expressions
Recall that the expression bn = x, b is a number multiplied n times to get the product x. Some simple situations are taking the squares and cubes of numbers. Take for example the following:
a. Squares of numbers:
(4)2 = (4)(4) = 16 x = 4, b = 16, n = 2
(-4)2 = (-4)(-4) = 16 x = -4, b = 16, n = 2
b. Cubes of numbers:
(2)3 = (2)(2)(2) = 8 x = 2, b = 8, n = 3
(-2)3 = (-2)(-2)(-2) = -8 x = -2, b = 8, n = 3
If we reverse the situation and find the value of b instead or the number multiplied n times to get x. This process shall be introduced to you in this section called radical expressions.
Using radical signs, such as [pic] are called radical expressions. Some simple radical expressions are square roots and cube roots. We shall use the radical symbol[pic]for square roots and [pic] for cube roots. This will give you a clearer understanding of perfect squares and perfect cubes. Continue reading the lessons.
Lesson 1
Identifying Expressions which are Perfect Squares
or Perfect Cubes
Perfect Squares:
A number is a perfect square if it is a product of a single number multiplied two times by itself.
Examples:
1. 16 is a perfect square because:
(4)(4) or (4)2 = 16 (-4)(-4) or (-4)2 = 16
2. [pic]is a perfect square because:
[pic]or [pic]= [pic] [pic]or [pic]= [pic]
3. 0.25 is a perfect square because:
(0.5)(0.5) or (0.5)2 = 0.25 (-0.5)(-0.5) or (-0.5)2 = 0.25
Perfect Cubes:
A number is a perfect cube if it is a product of a single number multiplied three times by itself.
Examples:
a. 33 or (3)(3)(3) = 27
b. (-5)3 or (-5)(-5)(-5) = -125
c. [pic]or [pic][pic][pic]= [pic]
Try this out
A. Identify if the following radical expressions are perfect squares.
1. 144 6. -125
2. 243 7. [pic]
3. 225 8. [pic]
4. 1296 9. 0.0008
5. 784 10. 0.025
B. Identify if the following radical expressions are perfect cubes.
1. 81 6. 343
2. -125 7. 500
3. 216 8. [pic]
4. 0.064 9. 1345
5. 1000 10. 729
C. Tell whether the following radical expressions are perfect squares or perfect cubes or neither.
1. 3.24 6. 6.4
2. 0.1849 7. 0.001
3. 7.84 8. 0.513
4. 44.1 9. 759
5. 15.21 10. 850
Lesson 2
Finding the Square Root or Cube Root of Radical Expressions
Square Roots:
Every positive number has a real number square roots. Negative numbers do not have real number square roots. The number zero has just one square root and that is 0 itself.
Consider the expression [pic]:
a. [pic]is a rational number, if x is a perfect square.
b. [pic]is an irrational number, if x is not a perfect square.
The radical expression[pic] do not have real square roots.
Examples:
1. [pic] is a rational number because 9 is a perfect square.
2. [pic] is an irrational number because 8 is not a perfect square.
3. [pic] is not a real number because no number squared will give you -64.
You should be careful not to confuse the idea of the negative square root of a number with the idea of a square root of a negative number.
-[pic] = -3 -[pic] can also be expressed as – ([pic]).
-[pic]= -5 -[pic] can be expressed as – ([pic]).
Now let us find square roots of numbers and identify those that do not qualify the given conditions.
Examples:
1. [pic]= 8 8 is a square root of 64 since (8)(8) = 64.
[pic]= -8 -8 is a square root of 64 since (-8)(-8) = 64
2. What is [pic]?
Answer: -25 does not have a real square root because there is no real number y such y2 = -25.
3. Find the two square roots of 100.
Answer: The square roots of 100 are 10 and –10 since, 102 = 100 and (-10)2 = 100.
4. The square root of [pic]= 39, you just cancel the exponent of 39 and the index.
Cube Roots:
In a similar way, If y is cubed and we get the value x, then y is the cube root of x. In symbol, y = [pic](read as y is the cube root of x ) if y3 = x.
Examples:
1. [pic] 4 is the cube root of 64 since (4) (4) (4) = 64.
2. [pic] –4 is the cube root of -64 since (-4) (-4) (-4) = -64.
3. [pic] [pic] is the cube root of [pic] since [pic][pic][pic] = [pic]
4. [pic] 35 is not a perfect cube because there is no number multiplied by itself 3 times will give you 35.
Try this out
A. Find the square roots of the following.
1. -[pic] 6. [pic]
2. [pic] 7. [pic]
3. [pic] 8. [pic]
4. [pic] 9. [pic]
5. [pic] 10. [pic]
B. Find the cube root of the following.
1. [pic] 6. [pic]
2. [pic] 7. [pic]
3. [pic] 8. [pic]
4. [pic] 9. [pic]
5. [pic] 10. [pic]
C. Find the indicated root of the following.
1. [pic] 6. [pic]
2. [pic] 7. [pic]
3. [pic] 8. [pic]
4. [pic] 9. [pic]
5. [pic] 10. [pic]
Lesson 3
Naming Two Rational Numbers Between Which a
Radical Expression Lies
Not all radicals have radicands which are perfect squares or perfect cubes. For example, [pic]etc. These are called irrational radicals.
In the case of these irrational radicals, the square roots are non-integers.
On the number line, the irrationals fill up the spaces between the rational numbers to compose the complete set of real numbers.
Example 1:
0 1 2 3 4
Example 2:
Some of the irrational radicals with their approximations are:
1. [pic] = 1.414… 2. [pic]= 1.732… 3. [pic]= 2.449…
Square roots of these numbers can be estimated by looking for the values between which the square root lies.
Examples:
1. Find two consecutive integers between which [pic]lies.
Solution: Since [pic]= 6 and [pic]= 7
Then [pic]
Therefore 6< [pic] < 7
Thus, [pic]lies between 6 and 7.
2. Find two consecutive integers between which [pic]lies.
Solution: Since [pic]= 10 and [pic]= 11
Then [pic]
Therefore 10 < [pic] < 11
Thus, [pic]lies between 10 and 11.
3. Find two consecutive integers between which [pic]lies.
Solution: Since [pic]= 2 and [pic]
Then [pic]
Therefore 2 < [pic]< 3
Thus, [pic]lies between 2 and 3.
We can find approximate square roots of irrational numbers through a calculator.
Most calculators with the “square root key” [pic] easily give the square root of a number.
To find the [pic], use the keystroke sequence the display is 8.660254.
You can also find approximate cube roots of irrational numbers by using a scientific calculator. Here’s how. Press key and key.
Examples:
1. To find [pic], press
2.1544347 will appear on display.
2. To find [pic], press
-2.1544347 will appear on display.
Try this out
A. Name two rational numbers between which each of the following radicals lie.
1. [pic] 6. [pic]
2. [pic] 7. [pic]
3. [pic] 8. [pic]
4. [pic] 9. [pic]
5. [pic] 10. [pic]
B. Name two rational numbers between which each of the following radicals lie.
1. [pic] 6. [pic]
2. [pic] 7. [pic]
3. [pic] 8. [pic]
4. [pic] 9. [pic]
5. [pic] 10. [pic]
C. Use your calculator to find an approximate value to two decimal places.
1. [pic] 6. [pic]
2. [pic] 7. [pic]
3. -[pic] 8. [pic]
4. -[pic] 9. [pic]
5. [pic] 10. [pic]
D. WHO IS THIS MATHEMATICIAN
This Polish mathematician was the first to use the symbol [pic]for square root. Born in 1489, he studied algebra at the University of Vienna between 1517 and 1521. He wrote Coss, The first German Algebra book in 1525.
To find out:
1. Find the answer to each number.
2. Write the letter under its matching number in the DECODER.
Let’s summarize
1. Perfect Squares:
A number is a perfect square if it is a product of a single number multiplied two times by itself.
2. Perfect Cubes:
A number is a perfect cube if it is a product of a single number multiplied three times by itself.
3. A radical expression is an expression using the radical sign, [pic].
4. Square Roots:
Consider the expression [pic]:
a. [pic]is a rational number, if x is a perfect square.
b. [pic]is an irrational number, if x is not a perfect square.
The radical expression[pic] do not have real square roots.
5. Cube Roots:
If y is cubed and we get the value x, then y is the cube root of x. In symbol, y = [pic](read as y is the cube root of x ) if y3 = x.
6. On the number line, the irrationals fill up the spaces between the rational numbers to compose the complete set of real numbers.
What have you learned
A. Find the indicated square roots of the following.
1. [pic] 6. [pic]
2. [pic] 7. [pic]
3. [pic] 8. [pic]
4. [pic] 9. [pic]
5. [pic] 10. [pic]
B. Find the cube roots of the following.
1. [pic] 6. [pic]
2. [pic] 7. [pic]
3. [pic] 8. [pic]
4. [pic] 9. [pic]
5. [pic] 10. [pic]
C. Name two rational numbers between which it lies.
1. [pic] 6. [pic]
2. [pic] 7. [pic]
3. [pic] 8. [pic]
4. [pic] 9. [pic]
5. [pic] 10. [pic]
Answer Key
How much do you Know
A.
1. 13
2. 20
3. 35
4. 1.5
5. 0.02
6. 0.009
7. 0.14
8. 1.3
9. [pic]
10. [pic]
B.
1. 3
2. –5
3. 7
4. 0.1
5. 0.4
6. [pic]
7. [pic]
8. [pic]
9. [pic]
10. [pic]
C.
1. 2 and 3
1. 3 and 4
2. 4 and 5
3. 5 and 6
4. 6 and 7
5. 4 and 5
6. 5 and 6
7. 6 and 7
8. 7 and 8
9. 9 and 10
Try this out
Lesson 1
A.
1. Perfect square
2. Not perfect square
3. Perfect square
4. Perfect square
5. Perfect square
6. Not perfect square
7. Not perfect square
8. Perfect square
9. Not perfect square
10. Not perfect square
B.
1. not perfect cube
2. Perfect cube
3. Perfect cube
4. Perfect cube
5. Perfect cube
6. Perfect cube
7. Not perfect cube
8. Perfect cube
9. Not perfect cube
10. Perfect cube
C.
1. Perfect square
2. Perfect square
3. Perfect square
4. Neither
5. Perfect square
6. Neither
7. Perfect cube
8. Perfect cube
9. Neither
10. Neither
Lesson 2
A.
1. –5
2. 15
3. 11
4. no real root
5. 0.15
6. 1.3
7. 0.04
8. [pic]
9. [pic]
10. [pic]
B.
1. 8
2. 2
3. –3
4. 5
5. 7
6. 15
7. 0.4
8. 0.5
9. –8
10. [pic]
C.
1. 50
2. 50
3. 5
4. –4
5. 7
6. 3
7. 2
8. 5
9. 78
10. 0.2
Lesson 3
A.
1. [pic]
2. [pic]
3. [pic]
4. [pic]
5. [pic]
6. [pic]
7. [pic]
8. [pic]
9. [pic]
10. [pic]
B.
1. [pic]
2. [pic]
3. [pic]
4. [pic]
5. [pic]
6. [pic]
7. -[pic]
8. [pic]
9. [pic]
10. [pic]
C.
1. 134.16
2. 70.71
3. –15.81
4. –18.97
5. 0.67
6. –2.93
7. 7.07
8. 8.41
9. 4.97
10. 6.07
D.
1. C
2. H
3. R
4. I
5. S
6. T
7. O
8. F
9. F
10. R
11. U
12. D
13. O
14. L
15. F
16. F
He is Christoff Rudolff
How much have you learned?
A.
1. 13
2. 21
3. 36
4. 28
5. 2.8
6. 0.02
7. [pic]
8. [pic]
9. [pic]
10. [pic]
B.
1. 3
2. –5
3. 7
4. 0.1
5. 0.4
6. [pic]
7. [pic]
8. [pic]
9. [pic]
10. [pic]
C.
1. 2 and 3
2. 4 and 5
3. 6 and 7
4. 9 and 10
5. 12 and 13
6. 16 and 17
7. 19 and 20
8. 13 and 14
9. 14 and 15
10. 10 and 11
-----------------------
Y
[pic]
[pic]
[pic]
(
(
(
[pic]
5
7
Shift
[pic]
[pic]
=
3
Shift
0
1
[pic]
=
3
Shift
(
0
1
[pic]
U
[pic]
C
[pic]
O
[pic]
I
[pic]
D
[pic]
H
[pic]
T
[pic]
F
[pic]
R
[pic]
L
[pic]
S
_____ _____ _____ _____ _____ _____ _____ _____ _____
[pic] 10 20 2 [pic] 4 [pic] 12 12
_____ _____ _____ _____ _____ _____ _____
20 14 [pic] [pic] 13 12 12
X
[pic]
................
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