9.7 Summary of key concepts

Elementary algebra Page 1 / 1

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 $\sqrt{x}$ ( [link] )

x

is a positive real number, then

\sqrt{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 $- \sqrt{x}$ ( [link] )

- \sqrt{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

\sqrt{x},

√ is called the radical sign .

x

is called the radicand .

\sqrt{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:

\sqrt{- 25}

is not meaningful and

\sqrt{- 25}

is not a real number

Simplifying square root expressions ( [link] )

a

is a nonnegative number, then

\sqrt{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.

\sqrt{2}, \sqrt{5}

and

\sqrt{10}

are irrational numbers

The product property ( [link] )

\begin{matrix} \sqrt{x y} & = & \sqrt{x} \end{matrix} \sqrt{y}

The quotient property ( [link] )

\begin{matrix} \sqrt{\frac{x}{y}} = \frac{\sqrt{x}}{\sqrt{y}} & y \neq 0 \end{matrix}

Be careful ( [link] )

\begin{array}{l} \sqrt{x + y} & \neq & \sqrt{x} + \sqrt{y} & (\sqrt{16 + 9} & \neq & \sqrt{16} + \sqrt{9}) \\ \sqrt{x - y} & \neq & \sqrt{x} - \sqrt{y} & (\sqrt{25 - 16} & \neq & \sqrt{25} - \sqrt{16}) \end{array}

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

perfect squares in the radicand,
fractions in the radicand, or
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.

\sqrt{x} \sqrt{y} = \sqrt{x y}

Simplify each square root, if necessary.
Perform the multiplication.
Simplify, if necessary.

Dividing square root expressions ( [link] )

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

\frac{\sqrt{x}}{\sqrt{y}} = \sqrt{\frac{x}{y}}

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

\begin{array}{l} a \sqrt{x} + b \sqrt{x} = (a + b) \sqrt{x} \\ a \sqrt{x} - b \sqrt{x} = (a - b) \sqrt{x} \end{array}

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] )

Isolate a radical.
Square both sides of the equation.
Simplify by combining like terms.
Repeat step 1 if radical are still present.
Obtain potential solution by solving the resulting non-square root equation.
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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9.7 Summary of key concepts

Summary of key concepts

Square root ( [link] )

Principal square root x ( [link] )

Secondary square root − x ( [link] )

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

Meaningful expressions ( [link] )

Simplifying square root expressions ( [link] )

Perfect squares ( [link] )

Irrational numbers ( [link] )

The product property ( [link] )

The quotient property ( [link] )

Be careful ( [link] )

Simplified form ( [link] )

Rationalizing the denominator ( [link] )

Multiplying square root expressions ( [link] )

Dividing square root expressions ( [link] )

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

Square root equation ( [link] )

Solving square root equations ( [link] )

Questions & Answers

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Principal square root $\sqrt{x}$ ( [link] )

Secondary square root $- \sqrt{x}$ ( [link] )