Module 2 - DEPED-LDN
(Effective Alternative Secondary Education)
MATHEMATICS II
MODULE 2
Radical Expressions
BUREAU OF SECONDARY EDUCATION
Department of Education
DepEd Complex, Meralco Avenue, Pasig City
Module 2
Radical Expressions
What this module is about
This module is about radical expressions and rational exponents. Some lessons include transforming radical expressions into exponential form and vice versa. You will also develop skills in simplifying radical expressions and expressions with rational exponents. Several activities are provided for you in this module.
What you are expected to learn
This module is designed for you to:
• define rational exponents.
• write expressions in radical or exponential form.
• simplify expressions involving rational exponents.
How much do you know
Express the following in radical form.
[pic] [pic]
Express the following in exponential form.
1. [pic] 6. [pic]
2. [pic] 7. [pic]
3. [pic] 8. [pic]
4. [pic] 9. [pic]
5. [pic] 10. [pic]
Change the indicated roots into radicals and evaluate.
[pic] [pic]
What you will do
Lesson 1
Write Expressions with Rational Exponents as
Radical Expressions and Vice Versa
In the radical expressions [pic] we shall call n the index. The exponent of x in this expression is 1. [pic] can be transformed as the rational expression [pic] and vice versa. Notice that in the rational expression [pic], the denominator n of the exponent is the index and the numerator 1 is the index of the radical expression [pic].
You can use this knowledge to write expressions involving rational exponents as radicals and vice versa.
Writing Rational Expressions in Radical Form:
If n is a positive integer, [pic].
Examples:
Write each rational expression in radical form.
1) [pic] = [pic]
2) [pic] = [pic]
3) [pic] = [pic]
If [pic] is any rational number where [pic] .
Examples:
Write each rational expression in radical form.
1) [pic]
2) [pic]
3) [pic]
4) [pic]
Examples:
Write each rational expression in radical form.
1) [pic] 3) [pic]
2) [pic] 4) [pic]
Solutions:
1) [pic]
2) [pic]
3) [pic]
4) [pic] or [pic]
Writing Radical Expressions in Exponential Form:
In writing radical expressions in exponential form such as [pic], where n is a positive integer, the index is the denominator while the exponent is the numerator.
Examples:
Write each radical expressions in exponential form.
1) [pic] 3) [pic]
2) [pic] 4) [pic]
Solutions:
1) [pic] = [pic]
2) [pic] = [pic]
3) [pic] = [pic]
4) [pic]= [pic]
Try this out
Write each radical expressions in radical form.
1) [pic] 6) [pic]
2) [pic] 7) [pic]
3) [pic] 8) [pic]
4) [pic] 9) [pic]
5) [pic] 10) [pic]
B. Write in exponential form
1. [pic] 6) [pic]
2) [pic] 7) [pic]
3) [pic] 8) [pic]
4) [pic] 9) [pic]
5) [pic] 10) [pic]
C. Math Integration
Where Is the Temple of Artemis?
The temple of Artemis is one of the seven wonders of the world. It was built mostly of marble around 550 B.C. in honor of a Greek goddess, Artemis.
In what country can this temple be found?
To answer, simplify the following radical expressions. Cross out each box that contains an answer. The remaining boxes will spell out the answer to the question.
1) [pic] = ___________
2) [pic] = ___________
3) -[pic] = ___________
4) [pic] = ___________
5) [pic] = ___________
6) [pic] = ___________
7) [pic] = ___________
8) [pic] = ___________
|G |T |E |U |G |R |E |
| | | | | | | |
|2 |9 |27 |1 |-8 |12 |8 |
|Y |P |K |E |C |T |Y |
| | | | | | | |
|[pic] |32 |4 |[pic] |16 |[pic] |[pic] |
Answer: _____ _____ _____ _____ _____ _____
Source: Math Journal
Volume X – Number 3
SY 2002-2003
Lesson 2
Simplifying Expressions Involving Rational Exponents
We shall use previous knowledge of transforming rational expressions to radical to simplify rational expressions.
Examples:
Express the given expression in radical form and then simplify.
1. [pic]
Solution:
a = 2
2) [pic]
3) [pic]
4) [pic]
5) [pic]
6) [pic]
For any real number a and positive integers m and n with n >1, [pic].
The two radical forms for [pic] are equivalent, and the choice of which form to use generally depends on whether we are evaluating numerical expressions or rewriting expressions containing variables in radical form.
Examples: Simplify expressions with rational exponents.
1) [pic]
= [pic] Simplify
= 27
Here, you take[pic], then cube the result. This will give you the answer 27.
2) [pic]
= [pic]
= [pic]
3) [pic]
= [pic]
= 4
Note that in example 1, you could also have evaluated the expression as
[pic] = 27.
This shows why we use [pic] for [pic] when evaluating numerical expressions. The numbers involved will be smaller and easier to work with
Try this out
Use the definition of [pic] to evaluate each expression.
1) [pic] 6) [pic]
2) [pic] 7) [pic]
3) [pic] 8) [pic]
4) [pic] 9) [pic]
5) [pic] 10) [pic]
Use the definition of [pic] to evaluate each expression.
1) [pic] 6) [pic]
2) [pic] 7) [pic]
3) [pic] 8) [pic]
4) [pic] 9) [pic]
5) [pic] 10) [pic]
Math Integration
The First Man to Orbit the Earth
In 1961, this Russian cosmonaut orbited the earth in a spaceship. Who was he?
To find out, evaluate the following. Then encircle the letter that corresponds to the correct answer. These letters will spell out the name of this Russian cosmonaut. Have fun!
1) [pic] Y. 12 Z. 14
2) [pic] O. 9 U. 13
3) [pic] Q 25 R. 5
4) [pic] E. 16 I. 6
5) [pic] G. 5 H. 25
6) [pic] A. 27 E. 9
7) [pic] F. –4 G. 4
8) [pic] A. 81 E. -81
9) [pic] R. –7 S. 7
10) [pic] E. –16 I. 16
11) [pic] N. [pic] P. [pic]
Answer:
1 2 3 4 5 6 7 8 9 10 11
Source: Math Journal – Volume XI-Number 2
SY 2003-2004
ISSN 0118-1211
Now you can extend your knowledge on rational exponent notation. Using the definition of negative exponents, you can write
[pic]
Examples:
Simplify each expression.
1) [pic]
2) [pic]
= [pic]
= [pic]
Try this out
Simplify each expression.
1) [pic] 6) [pic]
2) [pic] 7) [pic]
3) [pic] 8) [pic]
4) [pic] 9) [pic]
5) [pic] 10) [pic]
Let’s Summarize
As you mentioned earlier, you assume that all your previous exponent properties will continue to hold for rational exponents. These properties are restated here.
For any nonzero real numbers a and b and rational numbers m and n,
1) Product Rule [pic]
Quotient Rule [pic]
Power Rule [pic]
Product-power rule [pic]
Quotient-power rule [pic]
In general, [pic] if x and n are real numbers and n > 0.
For any real numbers a and positive integers m and n where n > 1 , [pic].
What have you learned
Express the following in radical form.
1) [pic] 6. [pic]
2) [pic] 7. [pic]
3) [pic] 8. [pic]
4) [pic] 9. [pic]
5) [pic] 10. [pic]
Express the following in exponential form.
1) [pic] 6) [pic]
2) [pic] 7) [pic]
3) [pic] 8) [pic]
4) [pic] 9) [pic]
5) [pic] 10) [pic]
Change the indicated roots into radicals and evaluate.
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
1) [pic]
2) [pic]
3) [pic]
4) [pic]
5) [pic]
6) [pic]
7) [pic]
8) [pic]
9) [pic]
10) [pic]
1) [pic]
2) [pic]
3) [pic]
4) [pic]
5) [pic]
6) [pic]
7) [pic]
8) [pic]
9) [pic]
10) [pic]
1) [pic]
2) [pic]
3) [pic]
4) [pic]
5) [pic]
6) [pic]
7) [pic]
8) [pic]
9) [pic]
10) [pic]
Try this out
Lesson 1
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) 2
2) 8
3) –8
4) 32
5) [pic]
6) [pic]
7) 27
8) 16
Answer: TURKEY
Lesson 2
1) 6
2) 10
3) 5
4) 8
5) 3
6) –4
7) 3
8) –2
9) [pic]
10) [pic]
B.
1) 9
2) 64
3) 16
4) 25
5) 4
6) [pic]
7) –27
8) [pic]
9) 729
10) -27
C. 1) 12 Y
2) 13 U
3) 5 R
4) 6 I
5) 5 G
6) 27 A
7) 4 G
8) 81 A
9) -7 R
10) 16 I
11) [pic] N
Negative Exponents
Try this out
1) [pic]
2) [pic]
3) [pic]
4) [pic]
5) [pic]
6) [pic]
7) [pic]
8) [pic]
9) [pic]
10) [pic]
What have you learned
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) [pic]
2) [pic]
3) [pic]
4) [pic]
5) [pic]
6) [pic]
7) [pic]
8) [pic]
9) [pic]
10) [pic]
-----------------------
Follow the same procedure as in example 1.
Change the negative exponent to positive exponent. Then simplify.
Y
Write in radical form. Get the cube root of –8, it is –2.
Then, you square it.
Express the given expression in radical form.
The denominator 4 is the index.
The numerator 3 is the exponent of the radical expression.
The fourth root of 16 is 2.
The fourth root of 81 is 3.
Multiply [pic] by itself three times or you take the cube of [pic]. [pic]
Write the rational expression as radical expression. The denominator 2, is the index. The numerator 3, is the exponent of 9 or the radical expression[pic].
32 1/5 is the fifth root of 32.
[pic] is not real number number
27 1/3 is the cube root of 27.
[pic]
Square both sides
In the equation a2 = 4, you can see that a is the number whose square is 4; that is; a is the principal square root of 4.
2 is the principal square root of 4
The denominator is the index. If the index is 2, there is no need of writing.
The denominator is 3 and so the index is 3.
The denominator is 4 and so the index is 4.
Here we use [pic], which is generally the preferred form in this situation.
The exponent applies to mn because of the parenthesis.
Note that the exponent applies only to the variable y.
Now the exponent applies to 2y because of the parenthesis.
The index is 5 and the exponent is 1.
The index is 5 and the exponent is 1.
The index is 4 and the exponent is 3.
The index is 2 and the exponent is 3. Five is not included in the parenthesis so the fractional exponent is only for y.
X
[pic]
................
................
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