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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The distinction between the principal square root of the number x and the secondary square root of the number x is made by explanation and by example. The simplification of the radical expressions that both involve and do not involve fractions is shown in many detailed examples; this is followed by an explanation of how and why radicals are eliminated from the denominator of a radical expression. Real-life applications of radical equations have been included, such as problems involving daily output, daily sales, electronic resonance frequency, and kinetic energy.Objectives of this module: understand the process used in adding and subtracting square roots, be able to add and subtract square roots.

Overview

  • The Logic Behind The Process
  • The Process

The logic behind the process

Now we will study methods of simplifying radical expressions such as

4 3 + 8 3 or 5 2 x 11 2 x + 4 ( 2 x + 1 )

The procedure for adding and subtracting square root expressions will become apparent if we think back to the procedure we used for simplifying polynomial expressions such as

4 x + 8 x or 5 a 11 a + 4 ( a + 1 )

The variables x and a are letters representing some unknown quantities (perhaps x represents 3 and a represents 2 x ). Combining like terms gives us

4 x + 8 x = 12 x or 4 3 + 8 3 = 12 3 and 5 a - 11 a + 4 ( a + 1 ) or 5 2 x - 11 2 x + 4 ( 2 x + 1 ) 5 a - 11 a + 4 a + 4 5 2 x - 11 2 x + 4 2 x + 4 - 2 a + 4 - 2 2 x + 4

The process

Let’s consider the expression 4 3 + 8 3 . There are two ways to look at the simplification process:

  1. We are asking, “How many square roots of 3 do we have?”

    4 3 means we have 4 “square roots of 3”

    8 3 means we have 8 “square roots of 3”

    Thus, altogether we have 12 “square roots of 3.”
  2. We can also use the idea of combining like terms. If we recall, the process of combining like terms is based on the distributive property

    4 x + 8 x = 12 x because 4 x + 8 x = ( 4 + 8 ) x = 12 x

    We could simplify 4 3 + 8 3 using the distributive property.

    4 3 + 8 3 = ( 4 + 8 ) 3 = 12 3

Both methods will give us the same result. The first method is probably a bit quicker, but keep in mind, however, that the process works because it is based on one of the basic rules of algebra, the distributive property of real numbers.

Sample set a

Simplify the following radical expressions.

4 32 + 5 2 . Simplify 32 . 4 16 · 2 + 5 2 = 4 16 2 + 5 2 = 4 · 4 2 + 5 2 = 16 2 + 5 2 = 21 2

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- 3 x 75 + 2 x 48 - x 27 . Simple each of the three radicals . = - 3 x 25 · 3 + 2 x 16 · 3 - x 9 · 3 = - 15 x 3 + 8 x 3 - 3 x 3 = ( - 15 x + 8 x - 3 x ) 3 = - 10 x 3

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5 a 24 a 3 - 7 54 a 5 + a 2 6 a + 6 a . Simplify each radical . = 5 a 4 · 6 · a 2 · a - 7 9 · 6 · a 4 · a + a 2 6 a + 6 a = 10 a 2 6 a - 21 a 2 6 a + a 2 6 a + 6 a = ( 10 a 2 - 21 a 2 + a 2 ) 6 a + 6 a = - 10 a 2 6 a + 6 a Factor out - 2 a . (This step is optional .) = - 2 a ( 5 a 6 a - 3 )

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Practice set a

Find each sum or difference.

7 84 x 12 189 x + 2 21 x

48 21 x

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4 x 54 x 3 + 36 x 2 + 3 24 x 5 3 x

18 x 2 6 x + 3 x

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Sample set b

3 + 8 3 8 . We'll rationalize the denominator by multiplying this fraction by 1 in the form 3 + 8 3 + 8 . 3 + 8 3 8 · 3 + 8 3 + 8 = ( 3 + 8 ) ( 3 + 8 ) 3 2 ( 8 ) 2 = 9 + 3 8 + 3 8 + 8 8 9 8 = 9 + 6 8 + 8 1 = 17 + 6 8 = 17 + 6 4 · 2 = 17 + 12 2

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2 + 7 4 - 3 . Rationalize the denominator by multiplying this fraction by 1 in the from 4 + 3 4 + 3 . 2 + 7 4 - 3 · 4 + 3 4 + 3 = ( 2 + 7 ) ( 4 + 3 ) 4 2 - ( 3 ) 2 = 8 + 2 3 + 4 7 + 21 16 - 3 = 8 + 2 3 + 4 7 + 21 13

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Practice set b

Simplify each by performing the indicated operation.

5 ( 6 4 )

30 4 5

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( 5 + 7 ) ( 2 + 8 )

3 10 + 3 14

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( 3 2 2 3 ) ( 4 3 + 8 )

8 6 12

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4 + 5 3 8

12 + 8 2 + 3 5 + 2 10

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Exercises

For the following problems, simplify each expression by performing the indicated operation.

3 6 12 6

15 6

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6 40 + 8 80

12 10 + 32 5

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5 x y 2 x y 3 3 y 2 2 x 3 y

2 x y 2 2 x y

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24 2 54 4 12

4 6 8 3

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8 20 9 125 + 10 180

5

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14 + 2 56 3 136

5 14 6 34

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4 a x 3 x + 2 3 a 2 x 3 + 7 3 a 2 x 3

13 a x 3 x

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3 b y 5 y + 4 5 b 2 y 3 2 5 b 2 y 3

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( 1 + 3 ) ( 2 3 )

1 + 3

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( 3 2 ) ( 4 2 )

7 ( 2 2 )

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( 2 + 5 ) ( 2 + 3 5 )

17 + 4 10

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( 2 6 3 ) ( 3 6 + 2 3 )

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( 4 5 2 3 ) ( 3 5 + 3 )

54 2 15

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( 3 8 2 2 ) ( 4 2 5 8 )

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( 12 + 5 3 ) ( 2 3 2 12 )

42

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( 2 7 ) 2

11 4 7

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( 2 + 5 x ) 2

4 + 4 5 x + 5 x

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( 8 6 b ) 2

64 16 6 b + 6 b

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( 3 y 7 y ) 2

9 y 2 6 y 7 y + 7 y

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( 2 + 5 ) ( 2 5 )

1

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( 6 + 7 ) ( 6 7 )

29

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( x + y ) ( x y )

x y

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4 6 + 2

2 ( 6 2 ) 17

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4 + 5 4 5

21 + 8 5 11

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8 3 2 + 18

16 + 2 3 + 24 2 3 6 14

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2 b 3 b 3 b + 2 b

2 6 5

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Exercises for review

( [link] ) Simplify ( x 5 y 3 x 2 y ) 5 .

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( [link] ) Simplify ( 8 x 3 y ) 2 ( x 2 y 3 ) 4 .

64 x 14 y 14

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( [link] ) Write ( x 1 ) 4 ( x 1 ) 7 so that only positive exponents appear.

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( [link] ) Simplify 27 x 5 y 10 z 3 .

3 x 2 y 5 z 3 x z

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( [link] ) Simplify 1 2 + x by rationalizing the denominator.

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Questions & Answers

what is math number
Tric Reply
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Shirleen Reply
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Adu
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
kinnecy Reply
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
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I rally confuse this number And equations too I need exactly help
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salma
Commplementary angles
Idrissa Reply
hello
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im all ears I need to learn
Sherica
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Tamia
hii
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salma
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opoku
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Ali
greetings from Iran
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salut. from Algeria
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hi
Nharnhar
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Kimberly Reply
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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