9.8: Summary of Key Concepts

Last updated
Save as PDF

Page ID: 49400

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

Summary of Key Concepts

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 and one negative. They opposites of each other.

Principal Square Root \(\sqrt{x}\)

If \(x\) 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 \(-\sqrt{x}\)

\(-\sqrt{x}\) 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

In the expression \(\sqrt{x}\),

\(\sqrt{}\) is called the radical sign.

\(x\) is called the radicand.

\(\sqrt{x}\) is called a radical.

The horizonal bar that appears attached to the radical sign, \(\sqrt{}\), is a grouping symbol that specifies the radicand.

Meaningful Expressions

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

If \(a\) is a nonnegative number, then

\(\sqrt{a^2} = a\)

Perfect Squares

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

Irrational Numbers

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

\(\sqrt{xy} = \sqrt{x}\sqrt{y}\)

The Quotient Property

\(\sqrt{\dfrac{x}{y}} = \dfrac{\sqrt{x}}{\sqrt{y}}\), \(y \not = 0\)

Be Careful

\(\begin{array}{flushleft}
\sqrt{x + y} &\not = \sqrt{x} \sqrt{y} & (\sqrt{16 + 9} &\not = \sqrt{16} + \sqrt{9})\\
\sqrt{x-y} &\not = \sqrt{x} - \sqrt{y} & (\sqrt{25-16} &\not = \sqrt{25} - \sqrt{16})
\end{array}\)

Simplified Form

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

Radicalizing the Denominator

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

Multiplying Square Root Expressions

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

\(\sqrt{x}\sqrt{y} = \sqrt{xy}\)

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

Dividing Square Root Expressions

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

\(\dfrac{\sqrt{x}}{\sqrt{y}} = \sqrt{\dfrac{x}{y}}\)

Addition and Subtraction of Square Root Expressions

\(a\sqrt{x} + b\sqrt{x} = (a+b)\sqrt{x}\)

\(a\sqrt{x} - b\sqrt{x} = (a-b)\sqrt{x}\)

Square Root Equation

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

Solving Square Root Equation

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.