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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 concept of square root, be able to distinguish between the principal and secondary square roots of a number, be able to relate square roots and meaningful expressions and to simplify a square root expression.

Overview

  • Square Roots
  • Principal and Secondary Square Roots
  • Meaningful Expressions
  • Simplifying Square Roots

Square roots

When we studied exponents in Section [link] , we noted that 4 2 = 16 and ( 4 ) 2 = 16. We can see that 16 is the square of both 4 and 4 . Since 16 comes from squaring 4 or 4 , 4 and 4 are called the square roots of 16. Thus 16 has two square roots, 4 and 4 . Notice that these two square roots are opposites of each other.

We can say that

Square root

The square root of a positive number x is a number such that when it is squared the number x results.

Every positive number has two square roots, one positive square root and one negative square root. Furthermore, the two square roots of a positive number are opposites of each other. The square root of 0 is 0.

Sample set a

The two square roots of 49 are 7 and −7 since

7 2 = 49 and ( 7 ) 2 = 49

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The two square roots of 49 64 are 7 8 and 7 8 since

( 7 8 ) 2 = 7 8 · 7 8 = 49 64 and ( 7 8 ) 2 = 7 8 · 7 8 = 49 64

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Practice set a

Name both square roots of each of the following numbers.

1 4

1 2 and  1 2

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9 16

3 4 and  3 4

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Principal and secondary square roots

There is a notation for distinguishing the positive square root of a number x from the negative square root of x .

Principal square root: x

If x is a positive real number, then

x represents the positive square root of x . The positive square root of a number is called the principal square root of the number.

Secondary square root: x

x represents the negative square root of x . The negative square root of a number is called the secondary square root of the number.

x indicates the secondary square root of x .

Radical sign, radicand, and radical

In the expression x ,

is called a radical sign .

x is called the radicand .

x is called a radical .

The horizontal bar that appears attached to the radical sign, , is a grouping symbol that specifies the radicand.

Because x and x are the two square root of x ,

( x ) ( x ) = x and ( x ) ( x ) = x

Sample set b

Write the principal and secondary square roots of each number.

9. Principal square root is  9 = 3. Secondary square root is 9 = 3.

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15. Principal square root is  15 . Secondary square root is 15 .

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Use a calculator to obtain a decimal approximation for the two square roots of 34. Round to two decimal places.
On the Calculater Type 34 Press x Display reads: 5.8309519 Round to 5.83.
Notice that the square root symbol on the calculator is . This means, of course, that a calculator will produce only the positive square root. We must supply the negative square root ourselves.

34 5.83 and 34 5.83
Note: The symbol ≈ means "approximately equal to."

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Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
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A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
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emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
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please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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what are the types of wave
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answer
Magreth
progressive wave
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Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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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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