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Wednesday 17th of April
Calculations with Negative Numbers
Solving Linear Equations
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Algebra Expressions
Evaluating Expressions and Solving Equations
Fraction rules
Factoring Quadratic Trinomials
Multiplying and Dividing Fractions
Dividing Decimals by Whole Numbers
Adding and Subtracting Radicals
Subtracting Fractions
Factoring Polynomials by Grouping
Slopes of Perpendicular Lines
Linear Equations
Roots - Radicals 1
Graph of a Line
Sum of the Roots of a Quadratic
Writing Linear Equations Using Slope and Point
Factoring Trinomials with Leading Coefficient 1
Writing Linear Equations Using Slope and Point
Simplifying Expressions with Negative Exponents
Solving Equations 3
Solving Quadratic Equations
Parent and Family Graphs
Collecting Like Terms
nth Roots
Power of a Quotient Property of Exponents
Adding and Subtracting Fractions
Solving Linear Systems of Equations by Elimination
The Quadratic Formula
Fractions and Mixed Numbers
Solving Rational Equations
Multiplying Special Binomials
Rounding Numbers
Factoring by Grouping
Polar Form of a Complex Number
Solving Quadratic Equations
Simplifying Complex Fractions
Common Logs
Operations on Signed Numbers
Multiplying Fractions in General
Dividing Polynomials
Higher Degrees and Variable Exponents
Solving Quadratic Inequalities with a Sign Graph
Writing a Rational Expression in Lowest Terms
Solving Quadratic Inequalities with a Sign Graph
Solving Linear Equations
The Square of a Binomial
Properties of Negative Exponents
Inverse Functions
Rotating an Ellipse
Multiplying Numbers
Linear Equations
Solving Equations with One Log Term
Combining Operations
The Ellipse
Straight Lines
Graphing Inequalities in Two Variables
Solving Trigonometric Equations
Adding and Subtracting Fractions
Simple Trinomials as Products of Binomials
Ratios and Proportions
Solving Equations
Multiplying and Dividing Fractions 2
Rational Numbers
Difference of Two Squares
Factoring Polynomials by Grouping
Solving Equations That Contain Rational Expressions
Solving Quadratic Equations
Dividing and Subtracting Rational Expressions
Square Roots and Real Numbers
Order of Operations
Solving Nonlinear Equations by Substitution
The Distance and Midpoint Formulas
Linear Equations
Graphing Using x- and y- Intercepts
Properties of Exponents
Solving Quadratic Equations
Solving One-Step Equations Using Algebra
Relatively Prime Numbers
Solving a Quadratic Inequality with Two Solutions
Operations on Radicals
Factoring a Difference of Two Squares
Straight Lines
Solving Quadratic Equations by Factoring
Graphing Logarithmic Functions
Simplifying Expressions Involving Variables
Adding Integers
Factoring Completely General Quadratic Trinomials
Using Patterns to Multiply Two Binomials
Adding and Subtracting Rational Expressions With Unlike Denominators
Rational Exponents
Horizontal and Vertical Lines
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Simplifying Expressions Involving Variables

When simplifying expressions involving rational exponents and variables, we must be careful to write equivalent expressions. For example, in the equation (x2)1/2 = x it looks as if we are correctly applying the power of a power rule. However, this statement is false if x is negative because the 1/2 power on the left-hand side indicates the positive square root of x2. For example, if x = -3, we get [(-3)2]1/2 = 91/2 = 3, which is not equal to -3. To write a simpler equivalent expression for (x2)1/2, we use absolute value as follows.

Square Root of x2

(x2)1/2 = | x | for any real number x.

Note that the equation (x2)1/2 = | x | is an identity. It is also necessary to use absolute value when writing identities for other even roots of expressions involving variables.

Example 1

Using absolute value symbols with exponents

Simplify each expression. Assume the variables represent any real numbers and use absolute value symbols as necessary.

a) (x8y4)1/4



a) Apply the power of a product rule to get the equation (x8y4)1/4 = x2y. The lefthand side is nonnegative for any choices of x and y, but the right-hand side is negative when y is negative. So for any real values of x and y we have (x8y4)1/4 = x2| y |.

b) Using the power of a quotient rule, we get

This equation is valid for every real number x, so no absolute value signs are used.

Helpful hint

We usually think of squaring and taking a square root as inverse functions, which they are as long as we stick to positive numbers.We can square 3 to get 9, and then find the square root of 9 to get 3— what we started with. We don’t get back to where we began if we start with -3.

Because there are no real even roots of negative numbers, the expressions a1/2, x-3/4, and y1/6 are not real numbers if the variables have negative values. To simplify matters, we sometimes assume the variables represent only positive numbers when we are working with expressions involving variables with rational exponents. That way we do not have to be concerned with undefined expressions and absolute value.

Example 2

Expressions involving variables with rational exponents

Use the rules of exponents to simplify the following. Write your answers with positive exponents. Assume all variables represent positive real numbers.


a) x2/3x4/3 = x6/3 Use the product rule to add the exponents.
  = x2 Reduce the exponent.
b) = a1/2 - 1/4 Use the quotient rule to subtract the exponents.
  = a1/4 Simplify.
c) (x1/2y -3)1/2 = (x1/2)1/2(y -3)1/2 Power of a product rule
  = x1/4y -3/2 Power of a power rule
  Definition of negative exponent

d) Because this expression is a negative power of a quotient, we can first find the reciprocal of the quotient, then apply the power of a power rule:

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