<< 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

how to study physic and understand
Ewa Reply
what is conservative force with examples
Moses
what is work
Fredrick Reply
the transfer of energy by a force that causes an object to be displaced; the product of the component of the force in the direction of the displacement and the magnitude of the displacement
AI-Robot
why is it from light to gravity
Esther Reply
difference between model and theory
Esther
Is the ship moving at a constant velocity?
Kamogelo Reply
The full note of modern physics
aluet Reply
introduction to applications of nuclear physics
aluet Reply
the explanation is not in full details
Moses Reply
I need more explanation or all about kinematics
Moses
yes
zephaniah
I need more explanation or all about nuclear physics
aluet
Show that the equal masses particles emarge from collision at right angle by making explicit used of fact that momentum is a vector quantity
Muhammad Reply
yh
Isaac
A wave is described by the function D(x,t)=(1.6cm) sin[(1.2cm^-1(x+6.8cm/st] what are:a.Amplitude b. wavelength c. wave number d. frequency e. period f. velocity of speed.
Majok Reply
what is frontier of physics
Somto Reply
A body is projected upward at an angle 45° 18minutes with the horizontal with an initial speed of 40km per second. In hoe many seconds will the body reach the ground then how far from the point of projection will it strike. At what angle will the horizontal will strike
Gufraan Reply
Suppose hydrogen and oxygen are diffusing through air. A small amount of each is released simultaneously. How much time passes before the hydrogen is 1.00 s ahead of the oxygen? Such differences in arrival times are used as an analytical tool in gas chromatography.
Ezekiel Reply
please explain
Samuel
what's the definition of physics
Mobolaji Reply
what is physics
Nangun Reply
the science concerned with describing the interactions of energy, matter, space, and time; it is especially interested in what fundamental mechanisms underlie every phenomenon
AI-Robot
what is isotopes
Nangun Reply
nuclei having the same Z and different N s
AI-Robot
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