Embed Size (px)
Citation preview
(Effective Alternative Secondary Education)
MATHEMATICS II
MODULE 4
Radical Expressions
BUREAU OF SECONDARY EDUCATIONDepartment of Education
DepEd Complex, Meralco Avenue, Pasig City
Y
X
Module 4Radical Expressions
What this module is about
This module is about addition and subtraction of radical expressions. Adding and subtracting radical expressions is very much similar to adding and subtracting similar terms of polynomials. Similar terms or like terms are those with the same literal parts or literal factors. Radicals are similar if they have the same index and the same radicands. For example: and 5 are similar radicals whose index is 2 and whose radicand is 3x.
What you are expected to learn
This module is designed for you to:
1. add and subtract radical expressions2. solve simple problems involving radical operations.
How much do you know Simplify and perform the necessary operations: 1. 5 + 6
2. 5 +
3. +
4. 2
5. 7
6. 7 - 5
7. 16 - 3
2
8.
9. 5 - 8 + 3
10. + 2 - 5
What will you do
Lesson 1
Addition of Radical Expressions
Radicals may be combined into simple radical expression by adding similar terms. The sum of two radicals cannot be simplified if the radicals have different indices or different radicands.
+ cannot be simplified (indices are different
+ cannot be simplified (radicands are different
It is important that the radicals are written in simplest form before adding and subtracting.
+ can further be simplified. = 3
= + 3 add
= 4
Examples:
Find the sum of the following radical expressions:
1. 4 + 5 + 6
Solution: Add coefficients and annex their common radical factor
= (4 + 5 + 6)
3
= 15
2. 6 + 3 + 2 + 5
Solution:
= (6 + 2 ) + (3 + 6 ) Group similar radicals.
= 8 + 9 Add the coefficients of similar radicals.
You cannot add the coefficients of different radicand.
3. 6 + 2
Solution:
= 6 + 2 Factor each radicand
= 6(3 ) + 2(5 ) Simplify
= 18 + 10 Add the coefficients and annextheir common radical factor
= 28
4.
Solution:
= Factor each radicand such that onefactor is a perfect cube.
= 2
= (2 + 3)
= 5
5.
Solution:
= Split the numerator and denominator
4
= Simplify and add the coefficients
=
6.
Solution:
= Split the numerator and denominator
and rationalize the denominator.
= Simplify
= Get the LCD. LCD = 6
= Add
=
7.
Solution:
= Split a7 such that the exponent is exactly divisible by the index, 2.
= 3a2 Simplify
= (3a2 + a)
5
Keep in mind that our basic approach to these problems has been to first put each term into simplest radical form before adding and /or subtracting.
Try this out Perform the indicated operations. A. 1. 4 + 5 2. 5 + 2 + 10
3. 16 + 8 + 4. 15 + 3 + 5 + 6
5.
B. 6. 2 + 3 7. + + 10 8. 3 + 9. + 2 + 10. 5 + C. 11. + 3 + 12. + +
13. + 2
14.
15. 2 + 5
D. What’s the Message?
6
Simplify the expression and find the correct answer in the box, then fill in the small box with the corresponding letter. Once you have filled in all the small boxes with the appropriate letter, darken all the blank spaces to separate the words in the secret message. What’s the message? Have fun!
1. E : 9. U:
2. N : 10. H: 2
3. T: 3 11. O: 3
4. S :6 12. L: 4
5. W: 3 13. R:
6. I: 14. C: 2
7. Q: 8 15. F:
8. Y: 16. V: 3
7
_ 100+5
_ 183
_ 83
27 53
_ 2/35
_153
__ 13
__14 + 3
__ 710
_ 83
_12 + 63
_12+63
_ _57+3
__ 710
__ 710
__ 5 10
_ _ 113+5
_ __7 +11
27 __ 710
__ 12 510
_ _ 82 +5
_ 25
_ 162
__44a5a
_ _2 +2 3
__ _ 410+ 3
8 7
_ 25
___5x211x
_ 43
53
__7y7y
_ 25
__ 710
_ _57 + 3
_ _66+52
_ 28x
__ 7y7y
_ 25
_ 2/35
_ _153+25
_ _2 + 3
_ 162
_ _57+3
_ _ 432+143
_ _57+3
___5x211x
LESSON 2
Subtraction of Radicals
Subtraction of radicals follows the rule in addition of real numbers. The operation just like in addition is only possible for similar radical expressions.
Examples:
Simplify the following:
1.
8
Solution:
= (5 – 2) Group their coefficients and subtract.
=
2.
Solution:
= Factor each radical in such a way thatone factor is a perfect square.
= Simplify
= Subtract
=
3.
Solution:
= 5x – 3x
= 2x
4.
Solution:
= Factor the radicals in such a way that one factor is a perfect square.
= 9, 4, x2 and y2 are perfect squares.
= Group the coefficients and subtract
9
= 1xy or xy
5. 2
Solution:
2 Factor the radicand in such a way that one factor is a perfect cube.
= 2 8 and 27 are perfect cubes hence, their cube root is 2 and 3 respectively.
= 2 Simplify
=
= -11
6.
Rename the expression such that the radicand can be expressed as a perfect cube and the exponents are exactly divisible by the index if possible then simplify.
Solution:
= 8, x3 and y3 are perfect cubes
=
= (2 - 2xy)
Subtraction of radical expressions with fraction.
7.
Solution:
10
= Split the numerator and denominator
and simplify.
= or
=
= Group the coefficients and simplify.
=
=
Addition and Subtraction of Radical Expressions
With the above knowledge, you can now perform a combination of operations.
1.
Solution:
= Group similar terms and perform the necessary operations.
=
Addition of dissimilar radicands is not allowed.2.
Solution:
= Factor in such a way that one factor is aperfect square.
= Simplify
11
= (3x – 2x + 5x) Perform the necessary operations.
= 6x
3. -
Solution:
-
= -
= 3a3 - a
= (3a3 – a)
4.
In this problem, no two radicands are identical and the radical expressions have indices 2, 3 and 4. The radical expression can be converted into a radical expression with index 2.
Solution:
=
Note that is the same as or = . You can now
complete the solution.
Since = then,
= Factor and simplify.
=
= Group similar radicands.
= By addition
12
Try This Out
Perform the indicated operations.
A. 1. - 2 2. 5 - 4 3. 3 - 7 4. 4 - 3 - 2
5. - 5 - 7 B. 6. -
7. 6 - + 2
8. -2 + 2 + 2
9. 5 + 3 - 3 10. - + 2 -
C. 11. 3 - 5 + 3
12. -
13. 2
14. 5
15. 3
D. Mental Math
Why are
13
Oysters greedy?
Perform and Simplify the following radicals. Write the letter in the box above its correct answer. Keep working and you will discover the answer the questions. H. - 18 + 4 - 5 E. 4 +
R. - 6 I. 7 +
T. 27 - 3 F. 3 - 2
S. 2 27 - 3 + Y. 4 27 + 5 - 2
L. A.
Let ‘s Summarize
When working with radicals, remember the following:
1. Put each radical into simplest form.
14
-23
-185
57
43+57
3 5 2
335 - 65
57
_-53
_-185
_57
1132
1132
-52
1010
-53
-185
Sums and difference of radical expressions can be simplified by applying the basic properties of real numbers.
The sum and difference of two radical expressions cannot be simplified if the radicals have different indices and different radicands.
2. Perform any indicated operations, if possible.
3. Make sure the final answer is also in simplest radical form.
What have you learned
A. Simplify each radical and perform the indicated operations.
1. 2 + 4
2. 5 -
3. 6 + 2 - 3
4. 2 - 5 -8
5. x – 3x + 5
6. 5 + 5 - 9
7. 3 - 2
8. + 5 - 7
9. 6 - 5 + 2
10.7 + 3 - 4
11. +
12. 6 +
13. 10 -
15
14. 20 -
15.
16
Answer Key
How much do you know
1. 11
2.
3.
4. 14
5. 2
6. -3
7. 4
8. 6 - 6 9. 11 10. 5
Lesson 1
Try this out
A. 1. 9
2. 17
3. 25
4. 29
5.
B. 6. 12 + 6
7. 41
8. 28
17
9. 13
10. 7
11. 15
12. 9
13.
14.
15.
C. What’s the Message?
1. E: 5 + 9. U: 8
2. N: 16 10. H: 7y
3. T: 53 11. O: 2
4. S: 7 12. L: 44a
5. W: 28 13. R: 5x2
6. I: 27 14. C: 12 + 6
7. Q: 18 15. F:
8. Y: + 2 16. V: 43 + 14
18
The Message: SUCCESS IS ONLY FOR THOSE WHO NEVER QUIT
Lesson 2
Try this out
A. 1. -
2.
3. -4
19
_ 183
Q
_ 83
U
27
I
53
T
__ 710
S
_ 83
U
_12 + 63
C
_12+63 +3
C
_ _57+3
E
__ 710
S
__ 710
S
27
I
_ 25
O
_162
N
__44a5a
L
_ _2 +2 3
Y
8 7
F
_ 25
O
___5x211x
R
53
T
__7y7y
H
_ 25
O
__ 710
S
_ _57 + 3
E
_28x
W
__7y7y
H
_ 25
O
_162
N
_ _57+3
E
_ _ 432+ 143
V
_ _57+3
E
___5x211x
R
__710
S
4. -
5. 2 - 10 - 14
B. 6. -4
7. 24 8. 0 9. 13
10. 0
C. 11. 7m
11. -7
12. 2
13.19 15. x
D. Mental Health
H. -18 E. 5
R. - 6 I. 10
T. -2 F. -5
S. -5 Y. 4 + 5
L. 11 A.
20
Why are Oysters greedy?
What have you learned
1. 6
2. 4
3. 5
4. -11
5. -2x + 5
6.
7. -5
8. 2 + 10 - 14
9. 4
10. 42 + 2
11.
12.
13. 30
21
T _-23
H _-185
E _57
Y _ _43+57
A3 5 2
R __ _335 - 65
E _57
S _-53
H _-185
E _57
L _1132
L _1132
F _-52
I _1010
S _-53
H _-185
14. 20 - or
15. 2
22
