<< Chapter < Page Chapter >> Page >

To simplify a square root expression that does not involve a fraction, we can use the following two rules:

Simplifying square roots without fractions

  1. If a factor of the radicand contains a variable with an even exponent, the square root is obtained by dividing the exponent by 2.
  2. If a factor of the radicand contains a variable with an odd exponent, the square root is obtained by first factoring the variable factor into two factors so that one has an even exponent and the other has an exponent of 1, then using the product property of square roots.

Sample set a

Simplify each square root.

a 4 .     The exponent is even: 4 2 = 2. The exponent on the square root is 2.

a 4 = a 2

Got questions? Get instant answers now!

a 6 b 10 .    Both exponents are even: 6 2 = 3 and 10 2 = 5. The exponent on the square root of a 6 is 3. The exponent on the square root if b 10 is 5.

a 6 b 10 = a 3 b 5

Got questions? Get instant answers now!

y 5 .    The exponent is odd: y 5 = y 4 y . Then

y 5 = y 4 y = y 4 y = y 2 y

Got questions? Get instant answers now!

36 a 7 b 11 c 20 = 6 2 a 6 a b 10 b c 20 a 7 = a 6 a , b 11 = b 10 b = 6 2 a 6 b 10 c 20 · a b by the commutative property of multiplication. = 6 2 a 6 b 10 c 20 a b by the product property of square roots. = 6 a 3 b 5 c 10 a b

Got questions? Get instant answers now!

49 x 8 y 3 ( a 1 ) 6 = 7 2 x 8 y 2 y ( a 1 ) 6 = 7 2 x 8 y 2 ( a 1 ) 6 y = 7 x 4 y ( a 1 ) 3 y

Got questions? Get instant answers now!

75 = 25 · 3 = 5 2 · 3 = 5 2 3 = 5 3

Got questions? Get instant answers now!

Practice set a

Simplify each square root.

81 a 12 b 6 c 38

9 a 6 b 3 c 19

Got questions? Get instant answers now!

144 x 4 y 80 ( b + 5 ) 16

12 x 2 y 40 ( b + 5 ) 8

Got questions? Get instant answers now!

w 7 z 3 k 13

w 3 z k 6 w z k

Got questions? Get instant answers now!

27 a 3 b 4 c 5 d 6

3 a b 2 c 2 d 3 3 a c

Got questions? Get instant answers now!

180 m 4 n 15 ( a 12 ) 15

6 m 2 n 7 ( a 12 ) 7 5 n ( a 12 )

Got questions? Get instant answers now!

Square roots involving fractions

A square root expression is in simplified form if there are

  1. no perfect squares in the radicand,
  2. no fractions in the radicand, or
  3. 3. no square root expressions in the denominator.

The square root expressions 5 a , 4 3 x y 5 , and 11 m 2 n a 4 2 x 2 are in simplified form.

The square root expressions 3 x 8 , 4 a 4 b 3 5 , and 2 y 3 x are not in simplified form.

Simplifying square roots with fractions

To simplify the square root expression x y ,

  1. Write the expression as x y using the rule x y = x y .
  2. Multiply the fraction by 1 in the form of y y .
  3. Simplify the remaining fraction, x y y .

Rationalizing the denominator

The process involved in step 2 is called rationalizing the denominator. This process removes square root expressions from the denominator using the fact that ( y ) ( y ) = y .

Sample set b

Simplify each square root.

3 5 = 3 5 = 3 5 · 5 5 = 15 5

Got questions? Get instant answers now!

9 8 = 9 8 = 9 8 · 8 8 = 3 8 8 = 3 4 · 2 8 = 3 4 2 8 = 3 · 2 2 8 = 3 2 4

Got questions? Get instant answers now!

k 2 m 3 = k 2 m 3 = k m 3 = k m 2 m = k m 2 m = k m m = k m m · m m = k m m m m = k m m · m = k m m 2

Got questions? Get instant answers now!

x 2 8 x + 16 = ( x 4 ) 2 = x 4

Got questions? Get instant answers now!

Practice set b

Simplify each square root.

32 a 5 b 7

4 a 2 2 a b b 4

Got questions? Get instant answers now!

Exercises

For the following problems, simplify each of the radical expressions.

100 x 5 y 11

10 x 2 y 5 x y

Got questions? Get instant answers now!

5 16 m 6 n 7

20 m 3 n 3 n

Got questions? Get instant answers now!

50 a 3 b 9

5 a b 4 2 a b

Got questions? Get instant answers now!

4 18 a 5 b 17

12 a 2 b 8 2 a b

Got questions? Get instant answers now!

4 75 a 4 b 6

20 a 2 b 3 3

Got questions? Get instant answers now!

9 a 2 b 3

3 a b b

Got questions? Get instant answers now!

m 6 n 8 p 12 q 20

m 3 n 4 p 6 q 10

Got questions? Get instant answers now!

49 x 2 y 5 z 9 25 a 3 b 11

7 x y 2 z 4 a b y z 5 a 2 b 6

Got questions? Get instant answers now!

( a 3 ) 4 ( a 1 ) 2

( a 3 ) 2 ( a 1 )

Got questions? Get instant answers now!

( b + 7 ) 8 ( b 7 ) 6

Got questions? Get instant answers now!

a 2 10 a + 25

( a 5 )

Got questions? Get instant answers now!

( a 2 2 a + 1 ) 4

( a 1 ) 4

Got questions? Get instant answers now!

Exercises for review

( [link] ) Solve the inequality 3 ( a + 2 ) 2 ( 3 a + 4 )

a 2 3

Got questions? Get instant answers now!

( [link] ) Graph the inequality 6 x 5 ( x + 1 ) 6.
A horizontal line with arrows on both ends.

Got questions? Get instant answers now!

( [link] ) Supply the missing words. When looking at a graph from left-to-right, lines with _______ slope rise, while lines with __________ slope fall.

positive; negative

Got questions? Get instant answers now!

( [link] ) Simplify the complex fraction 5 + 1 x 5 1 x .

Got questions? Get instant answers now!

( [link] ) Simplify 121 x 4 w 6 z 8 by removing the radical sign.

11 x 2 w 3 z 4

Got questions? Get instant answers now!

Questions & Answers

what does mean opportunity cost?
Aster Reply
what is poetive effect of population growth
Solomon Reply
what is inflation
Nasir Reply
what is demand
Eleni
what is economics
IMLAN Reply
economics theory describes individual behavior as the result of a process of optimization under constraints the objective to be reached being determined by
Kalkidan
Economics is a branch of social science that deal with How to wise use of resource ,s
Kassie
need
WARKISA
Economic Needs: In economics, needs are goods or services that are necessary for maintaining a certain standard of living. This includes things like healthcare, education, and transportation.
Kalkidan
What is demand and supply
EMPEROR Reply
deman means?
Alex
what is supply?
Alex
ex play supply?
Alex
Money market is a branch or segment of financial market where short-term debt instruments are traded upon. The instruments in this market includes Treasury bills, Bonds, Commercial Papers, Call money among other.
murana Reply
good
Kayode
what is money market
umar Reply
Examine the distinction between theory of comparative cost Advantage and theory of factor proportion
Fatima Reply
What is inflation
Bright Reply
a general and ongoing rise in the level of prices in an economy
AI-Robot
What are the factors that affect demand for a commodity
Florence Reply
price
Kenu
differentiate between demand and supply giving examples
Lambiv Reply
differentiated between demand and supply using examples
Lambiv
what is labour ?
Lambiv
how will I do?
Venny Reply
how is the graph works?I don't fully understand
Rezat Reply
information
Eliyee
devaluation
Eliyee
t
WARKISA
hi guys good evening to all
Lambiv
multiple choice question
Aster Reply
appreciation
Eliyee
explain perfect market
Lindiwe Reply
In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,
Ezea
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask