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9.7 Summary of key concepts

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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.This module presents a summary of the key concepts of the chapter "Roots, Radicals, and Square Root Equations".

Summary of key concepts

Square root ( [link] )

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 and one negative. They are opposites of each other.

Principal square root x ( [link] )

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 ( [link] )

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

Radical sign, radicand; and radical ( [link] )

In the expression x ,
√ is called the 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.

Meaningful expressions ( [link] )

A radical expression will only be meaningful if the radicand (the expression under the radical sign) is not negative:

− 25 is not meaningful    and     − 25 is not a real number

Simplifying square root expressions ( [link] )

If a is a nonnegative number, then

a 2 = a

Perfect squares ( [link] )

Real numbers that are squares of rational numbers are called perfect squares .

Irrational numbers ( [link] )

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

2 , 5   and   10 are irrational numbers

The product property ( [link] )

x y = x y

The quotient property ( [link] )

x y = x y y ≠ 0

Be careful ( [link] )

x + y ≠ x + y ( 16 + 9 ≠ 16 + 9 ) x − y ≠ x − y ( 25 − 16 ≠ 25 − 16 )

Simplified form ( [link] )

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

A square root involving a fraction is in simplified form if there are no
  1. perfect squares in the radicand,
  2. fractions in the radicand, or
  3. square root expressions in the denominator

Rationalizing the denominator ( [link] )

The process of eliminating radicals from the denominator is called rationalizing the denominator .

Multiplying square root expressions ( [link] )

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

x y = x y
  1. Simplify each square root, if necessary.
  2. Perform the multiplication.
  3. Simplify, if necessary.

Dividing square root expressions ( [link] )

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

x y = x y

Addition and subtraction of square root expressions ( [link] )

a x + b x = ( a + b ) x a x − b x = ( a − b ) x

Square root equation ( [link] )

A square root equation is an equation that contains a variable under a square root radical sign.

Solving square root equations ( [link] )

  1. Isolate a radical.
  2. Square both sides of the equation.
  3. Simplify by combining like terms.
  4. Repeat step 1 if radical are still present.
  5. Obtain potential solution by solving the resulting non-square root equation.
  6. Check potential solutions by substitution.

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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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