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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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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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