(Effective Alternative Secondary Education)(Effective Alternative Secondary Education)

MATHEMATICS II

MODULE 2

Radical Expressions

BUREAU OF SECONDARY EDUCATIONDepartment of Education

DepEd Complex, Meralco Avenue, Pasig City

Module 21

Y

X

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.

Express the following in exponential form.

1. 6.

5

2. 7. 3. 8. 4. 9. 5. 10. Change the indicated roots into radicals and evaluate.

What you will do

Lesson 1

Write Expressions with Rational Exponents as Radical Expressions and Vice Versa

In the radical expressions we shall call n the index. The exponent of x in this expression is 1. can be transformed as the rational

expression and vice versa. Notice that in the rational expression , the

denominator n of the exponent is the index and the numerator 1 is the index of the radical expression .

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, .

Examples:

5

The denominator is the index. If the index is 2, there is no need of writing.

Write each rational expression in radical form.

1) =

2) =

3) =

If is any rational number where .

Examples:

Write each rational expression in radical form.

1)

2)

3)

4)

Examples:

Write each rational expression in radical form.

1) 3)

2) 4)

Solutions:

1)

2)

3)

5

The denominator is 3 and so the index is 3.

The denominator is 4 and so the index is 4.

Here we use , 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.

4) or

Writing Radical Expressions in Exponential Form:

In writing radical expressions in exponential form such as ,

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) 3) 2) 4)

Solutions:

1) =

2) =

3) =

4) =

Try this out

Write each radical expressions in radical form.

1) 6)

2) 7)

3) 8)

4) 9)

5) 10)

B. Write in exponential form

1. 6) 2) 7) 3) 8)

5

The index is 5 and the exponent is 1.

The index is 4 and the exponent is 3.

The index is 5 and the exponent is 1.

The index is 2 and the exponent is 3. Five is not included in the parenthesis so the fractional exponent is only for y.

4) 9) 5) 10)

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) = ___________

2) = ___________3) - = ___________

4) = ___________

5) = ___________

6) = ___________

7) = ___________

8) = ___________

G

2

T

9

E

27

U

1

G

-8

R

12

E

8

Y P

32

K

4

E C

16

T Y

5

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

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.

Solution:

a = 2

2)

3)

4)

5

27 1/3 is the cube root of 27.

is not real number number

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 .

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 by itself three times or you

take the cube of .

5)

6)

For any real number a and positive integers m and n with n >1,

.

The two radical forms for 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)

= Simplify

= 27

Here, you take , then cube the result. This will give you the answer 27.

2)

=

=

3)

=

= 4

5

32 1/5 is the fifth root of 32.

Write in radical form. Get the cube root of –8, it is –2.

Then, you square it.

Note that in example 1, you could also have evaluated the expression as

= 27.

This shows why we use for when evaluating numerical

expressions. The numbers involved will be smaller and easier to work with

Try this out

Use the definition of to evaluate each expression.

1) 6)

2) 7)

3) 8)

4) 9)

5) 10)

Use the definition of to evaluate each expression.

1) 6)

2) 7)

3) 8)

4) 9)

5) 10)

Math Integration

The First Man to Orbit the Earth

In 1961, this Russian cosmonaut orbited the earth in a spaceship. Who was he?

5

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) Y. 12 Z. 14

2) O. 9 U. 13

3) Q 25 R. 5

4) E. 16 I. 6

5) G. 5 H. 25

6) A. 27 E. 9

7) F. –4 G. 4

8) A. 81 E. -81

9) R. –7 S. 7

10) E. –16 I. 16

11) N. P.

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

Examples:

Simplify each expression.

5

Change the negative exponent to positive exponent. Then simplify.

1)

2)

=

=

Try this out Simplify each expression.

1) 6)

2) 7)

3) 8)

4) 9)

5) 10)

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

5

Follow the same procedure as in example 1.

Quotient Rule

Power Rule Product-power rule

Quotient-power rule

In general, if x and n are real numbers and n > 0.

For any real numbers a and positive integers m and n where n > 1 ,

.

What have you learned

Express the following in radical form.

1) 6.

2) 7.

3) 8.

4) 9.

5) 10.

Express the following in exponential form.

1) 6) 2) 7) 3) 8)

4) 9)

5) 10) Change the indicated roots into radicals and evaluate.

1) 6)

2) 7)

5

3) 8)

4) 9)

5) 10)

\ Answer KeyHow much do you know

1) 2) 3)

4)

5)

6)

7)

8)

9) 10)

1)

2)

3)

4)

5)

6)

7)

8)

9)

5

10)

1) 2) 3) 4)

5) 6) 7) 8) 9) 10)

Try this outLesson 1

A. 1)

2)

3)

4)

5)

6) 7) 8)

9) 10)

B. 1)

2)

3)

4) 5)

6)

7)

8)

5

9)

10) C. 1) 2 2) 8 3) –8 4) 32

5)

6)

7) 27 8) 16

Answer: TURKEYLesson 2

1) 62) 103) 54) 85) 36) –47) 38) –2

9)

10)

B. 1) 92) 643) 164) 255) 4

6)

7) –27

8)

9) 72910)-27

C. 1) 12 Y2) 13 U3) 5 R4) 6 I

5

5) 5 G6) 27 A7) 4 G8) 81 A9) -7 R10) 16 I

11) N

Negative ExponentsTry this out

1)

2)

3)

4)

5)

6)

7)

8)

9)

10)

What have you learned A. 1) 2) 3) 4) 5) 6) 7) 8) 9) 10)

5

B. 1)

2)

3)

4)

5)

6)

7)

8)

9)

10)

C. 1)

2)

3) 4) 5)

6)

7)

8)

9)

10)

5