<< 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

calculate molarity of NaOH solution when 25.0ml of NaOH titrated with 27.2ml of 0.2m H2SO4
Gasin Reply
what's Thermochemistry
rhoda Reply
the study of the heat energy which is associated with chemical reactions
Kaddija
How was CH4 and o2 was able to produce (Co2)and (H2o
Edafe Reply
explain please
Victory
First twenty elements with their valences
Martine Reply
what is chemistry
asue Reply
what is atom
asue
what is the best way to define periodic table for jamb
Damilola Reply
what is the change of matter from one state to another
Elijah Reply
what is isolation of organic compounds
IKyernum Reply
what is atomic radius
ThankGod Reply
Read Chapter 6, section 5
Dr
Read Chapter 6, section 5
Kareem
Atomic radius is the radius of the atom and is also called the orbital radius
Kareem
atomic radius is the distance between the nucleus of an atom and its valence shell
Amos
Read Chapter 6, section 5
paulino
Bohr's model of the theory atom
Ayom Reply
is there a question?
Dr
when a gas is compressed why it becomes hot?
ATOMIC
It has no oxygen then
Goldyei
read the chapter on thermochemistry...the sections on "PV" work and the First Law of Thermodynamics should help..
Dr
Which element react with water
Mukthar Reply
Mgo
Ibeh
an increase in the pressure of a gas results in the decrease of its
Valentina Reply
definition of the periodic table
Cosmos Reply
What is the lkenes
Da Reply
what were atoms composed of?
Moses Reply
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