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Absolute Values
Solving Two-Step Equations Algebraically
Multiplying Monomials
Factoring Trinomials
Solving Quadratic Equations
Power Functions and Transformations
Composition of Functions
Rational Inequalities
Equations of Lines
Graphing Logarithmic Functions
Elimination Using Multiplication
Multiplying Large Numbers
Multiplying by 11
Graphing Absolute Value Inequalities
Polynomials
The Discriminant
Reducing Numerical Fractions to Simplest Form
Addition of Algebraic Fractions
Graphing Inequalities in Two Variables
Adding and Subtracting Rational Expressions with Unlike Denominators
Multiplying Binomials
Graphing Linear Inequalities
Properties of Numbers and Definitions
Factoring Trinomials
Relatively Prime Numbers
Point
Inequalities
Rotating a Hyperbola
Writing Algebraic Expressions
Quadratic and Power Inequalities
Solving Quadratic Equations by Completing the Square
BEDMAS & Fractions
Solving Absolute Value Equations
Writing Linear Equations in Slope-Intercept Form
Adding and Subtracting Rational Expressions with Different Denominators
Reducing Rational Expressions
Solving Absolute Value Equations
Equations of a Line - Slope-intercept form
Adding and Subtracting Rational Expressions with Unlike Denominators
Solving Equations with a Fractional Exponent
Simple Trinomials as Products of Binomials
Equivalent Fractions
Multiplying Polynomials
Slope
Graphing Equations in Three Variables
Properties of Exponents
Graphing Linear Inequalities
Solving Cubic Equations by Factoring
Adding and Subtracting Fractions
Multiplying Whole Numbers
Straight Lines
Solving Absolute Value Equations
Solving Nonlinear Equations
Factoring Polynomials by Finding the Greatest Common Factor
Logarithms
Algebraic Expressions Containing Radicals 1
Addition Property of Equality
Three special types of lines
Quadratic Inequalities That Cannot Be Factored
Adding and Subtracting Fractions
Coordinate System
Solving Equations
Factoring Polynomials
Solving Quadratic Equations
Multiplying Radical Expressions
Solving Quadratic Equations Using the Square Root Property
The Slope of a Line
Square Roots
Adding Polynomials
Arithmetic with Positive and Negative Numbers
Solving Equations
Powers and Roots of Complex Numbers
Adding, Subtracting and Finding Least Common Denominators
What the Factored Form of a Quadratic can tell you about the graph
Plotting a Point
Solving Equations with Variables on Each Side
Finding the GCF of a Set of Monomials
Completing the Square
Solving Equations with Radicals and Exponents
Solving Systems of Equations By Substitution
Adding and Subtracting Rational Expressions
Percents
Laws of Exponents and Dividing Monomials
Factoring Special Quadratic Polynomials
Radicals
Solving Quadratic Equations by Completing the Square
Reducing Numerical Fractions to Simplest Form
Factoring Trinomials
Writing Decimals as Fractions
Using the Rules of Exponents
Evaluating the Quadratic Formula
Rationalizing the Denominator
Multiplication by 429
Writing Linear Equations in Point-Slope Form
Multiplying Radicals
Dividing Polynomials by Monomials
Factoring Trinomials
Introduction to Fractions
Square Roots
   
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Algebraic Expressions Containing Radicals

A radical or root is sort of like the reverse of a power. We’ve already covered methods for working with radicals involving just numbers in some detail in a previous section of these notes. In this document (and four or so that follow), we extend methods already illustrated for radicals containing only numbers to radicals which contain symbols as well. Algebraic radicals are still indicated by the symbol .

The meaning of this symbol is given by the condition

That is, by , we mean the quantity (or expression) whose power is equal to x.

Remarks

(i) When n = 2, the small superscript is usually omitted, so

We call the “square root of x” because

(ii) The superscript n in is called the order of the radical or root. The general basic algebraic properties of roots do not depend on the order of the root, but the actual detailed symbolic manipulations required to work with higher order roots get more and more complicated as the order gets higher. It isn’t difficult to extend the methods we will illustrate here to expressions containing higher order roots if you understand the basic principles. However, we will not take the space to do so.

(iii) You are probably aware that roots or radicals can be written in equivalent exponential form:

and

Simplifying products or quotients of roots may be eased in some cases if the equivalent exponential forms are used. However, often (and always when sums or differences of radicals are present), conversion from radical to exponential notation will not likely be very helpful and may actually make simplifying the expression much more difficult.

Radicals are closely related to the operation of multiplication. This means that there are some simple properties for multiplying or factoring radicals. Specifically

and

There are no simple rules for breaking up radicals of sums or for combining sums of radicals! We will emphasize this over and over again in the notes to follow, but to alert you to a very common error, we state here that you can never do something like

or

When written specifically for square roots, what we have just said is that

and

but

in general. Here, x and y represent any non-negative number or valid mathematical expression that will evaluate to a non-negative value in the case of square roots.

(v) Radicals arise in technical applications because powers occur in technical applications. So, if you have squares of symbols appearing in formulas, solving problems based on such a formula will sometimes involve square roots. A very simple example is the following. The area, A, of a square whose sides have length s is given by

But then, if we know the area of the square and we need to compute the lengths of its sides, we must use the formula

Why? Because if we need to square s to get the value of A, then s must be the quantity whose square is A – that is, s is the square root of A.

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Tuesday 12th of December