Math 1
Notes on Radicals & Variables
Name_________________________
Standards Addresssed: MM1A2.
Students will simplify and operate with radical expressions, polynomials, and rational expressions.
a. Simplify algebraic and numeric expressions involving square root.
b. Perform operations with square roots.
--------------------------------------------------------------------------------------------------------------------Before simplifying radicals involving variables, let's simplify some expressions in which a
variable quantity is multiplied by itself.
Simplify.
2. a 2  a 2
1. y  y
3. b5  b5
4. c11  c11
--------------------------------------------------------------------------------------------------------------------Reminder - when trying to evaluate 25 , one asks the question, "What positive number times
itself equals 25?" Of course, the answer is 5.
Consider
y 2 . In order to simplify this expression, ask a similar question, "What variable
quantity times itself equals y 2 ?" If necessary, look at the questions above.
Therefore, what is
y2 ?
Use this same thought process and the answers above to simplify the following:
5.
a4
6.
b10
7.
c 22
8.
d 18
--------------------------------------------------------------------------------------------------------------------For Questions 5-8, examine the exponents of the radicands in the question. Then, look at the
exponents of the answers. In your own words, state the pattern of how to take the exponent from
the original question and get the exponent of the answer.
--------------------------------------------------------------------------------------------------------------------Another relationship between Questions 5-8 is that all of the exponents in the radicands are even.
When the exponents are odd, we have to do something slightly different.
Consider
m7 .
Let's use the pattern established above and divide the exponent (7) by 2. How many times does 2
go into 7? What is the remainder?
Therefore, the resulting expression is m3 m .
As you can see, when the exponent in the radicand is odd, there will be one variable that remains
in the radicand.
Simplify. If there are two variables in the radicand, deal with each one separately.
9.
n13
10.
r 21
11.
s 4 w10
x15 y 2
12.
--------------------------------------------------------------------------------------------------------------------Occasionally, positive numbers appear with variables in the radicands. If this happens, just deal
with the positive number by using the rules taught last week, and then deal with the variables.
13.
4x6
14.
81y 23
15.
20k 3
--------------------------------------------------------------------------------------------------------------------When adding, subtracting, or multiplying, just follow the rules taught previously for radicals and
variables.
16. a 2 b3  a b
17.
49 x5  3x2 x
--------------------------------------------------------------------------------------------------------------------Homework on Radicals and Variables
Simplify completely. Assume all variables represent positive numbers.
1.
d2
2.
c4
3.
a11
4.
b5
5.
c9 d 10
6.
j15 k
7.
4x6
8.
100 y 7
12.
c 200
9. 2 12z
10. 6a 2 9a 2
11. 12b 18a 4b
13. 5 a3  a a
14. 10b2  9b4
15. 16 x2 x  25x6  81x5
16. 2c d  4c3d 2
17. 3 y y  2 y y
18. 9 x3  2 x x
19. 4r 3 rs  2r 2 s
20. (10u u )(2u u )(3u w )
--------------------------------------------------------------------------------------------------------------------1. d
2. c 2
3. a5 a
4. b 2 b
5. c 4 d 5 c
6. j 7 jk
7. 2x 3
8. 10y3 y
9. 4 3z
10. 18a 3
11. 36a 2b 2b
15. 7 x 2 x  5 x3
18. 18x 3
12. c100
13. 6a a
16. 2c d  2cd c
19. 8r 5 s r
14. 7b 2
17. 6 y 3
20. 60u 4 w