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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: be able to identify a perfect square, be familiar with the product and quotient properties of square roots, be able to simplify square roots involving and not involving fractions.

Overview

  • Perfect Squares
  • The Product Property of Square Roots
  • The Quotient Property of Square Roots
  • Square Roots Not Involving Fractions
  • Square Roots Involving Fractions

To begin our study of the process of simplifying a square root expression, we must note three facts: one fact concerning perfect squares and two concerning properties of square roots.

Perfect squares

Perfect squares

Real numbers that are squares of rational numbers are called perfect squares. The numbers 25 and 1 4 are examples of perfect squares since 25 = 5 2 and 1 4 = ( 1 2 ) 2 , and 5 and 1 2 are rational numbers. The number 2 is not a perfect square since 2 = ( 2 ) 2 and 2 is not a rational number.

Although we will not make a detailed study of irrational numbers, we will make the following observation:

Any indicated square root whose radicand is not a perfect square is an irrational number.

The numbers 6 , 15 , and 3 4 are each irrational since each radicand ( 6 , 15 , 3 4 ) is not a perfect square.

The product property of square roots

Notice that

9 · 4 = 36 = 6      and
9 4 = 3 · 2 = 6

Since both 9 · 4 and 9 4 equal 6, it must be that

9 · 4 = 9 4

The product property x y = x y

This suggests that in general, if x and y are positive real numbers,

x y = x y

The square root of the product is the product of the square roots.

The quotient property of square roots

We can suggest a similar rule for quotients. Notice that

36 4 = 9 = 3      and
36 4 = 6 2 = 3

Since both 36 4 and 36 4 equal 3, it must be that

36 4 = 36 4

The quotient property x y = x y

This suggests that in general, if x and y are positive real numbers,

x y = x y ,       y 0

The square root of the quotient is the quotient of the square roots.

CAUTION
It is extremely important to remember that

x + y x + y or x y x y

For example, notice that 16 + 9 = 25 = 5 , but 16 + 9 = 4 + 3 = 7.

We shall study the process of simplifying a square root expression by distinguishing between two types of square roots: square roots not involving a fraction and square roots involving a fraction.

Square roots not involving fractions

A square root that does not involve fractions is in simplified form if there are no perfect square in the radicand.

The square roots x , a b , 5 m n , 2 ( a + 5 ) are in simplified form since none of the radicands contains a perfect square.

The square roots x 2 , a 3 = a 2 a are not in simplified form since each radicand contains a perfect square.

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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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