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Exponential Decay
Negative Exponents
Multiplying and Dividing Fractions 4
Evaluating Expressions Involving Fractions
The Cartesian Coordinate System
Adding and Subtracting Fractions with Like Denominators
Solving Absolute Value Inequalities
Multiplying Special Polynomials
FOIL Method
Inequalities
Solving Systems of Equations by Graphing
Graphing Compound Inequalities
Solving Quadratic Equations by Completing the Square
Addition Property of Equality
Square Roots
Adding and Subtracting Fractions
The Distance Formula
Graphing Logarithmic Functions
Fractions
Dividing Mixed Numbers
Evaluating Polynomials
Power of a Product Property of Exponents
Terminology of Algebraic Expressions
Adding and Subtracting Rational Expressions with Identical Denominators
Solving Exponential Equations
Factoring The Difference of 2 Squares
Changing Fractions to Decimals
Solving Linear Equations
Using Patterns to Multiply Two Binomials
Completing the Square
Roots of Complex Numbers
Methods for Solving Quadratic Equations
Conics in Standard Form
Solving Quadratic Equations by Using the Quadratic Formula
Simplifying Fractions 2
Exponential Notation
Exponential Growth
The Cartesian Plane
Graphing Linear Functions
The Slope of a Line
Finding Cube Roots of Large Numbers
Rotating Axes
Common Mistakes With Percents
Solving an Equation That Contains a Square Root
Rational Equations
Properties of Common Logs
Composition of Functions
Using Percent Equations
Solving Inequalities
Properties of Exponents
Graphing Quadratic Functions
Factoring a Polynomial by Finding the GCF
The Rectangular Coordinate System
Adding and Subtracting Fractions
Multiplying and Dividing Rational Expressions
Improper Fractions and Mixed Numbers
Properties of Exponents
Complex Solutions of Quadratic Equations
Solving Nonlinear Equations by Factoring
Solving Quadratic Equations by Factoring
Least Common Multiples
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Solving Exponential Equations
Solving Linear Equations
Multiplication Property of Equality
Multiplying Mixed Numbers
Multiplying Fractions
Reducing a Rational Expression to Lowest Terms
Literal Numbers
Factoring Trinomials
Logarithmic Functions
Adding Fractions with Unlike Denominators
Simplifying Square Roots
Adding Fractions
Equations Quadratic in Form
Dividing Rational Expressions
Slopes of Parallel Lines
Simplifying Cube Roots That Contain Variables
Functions and Graphs
Complex Numbers
Multiplying and Dividing Fractions 1
Composition of Functions
Intercepts of a Line
Powers
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Multiplying Two Numbers with the same Tens Digit and whose Ones Digits add up to 10
Factoring Trinomials
Exponents and Polynomials
Decimals and their Equivalent Fractions
Negative Integer Exponents
Adding and Subtracting Mixed Numbers
Solving Quadratic Equations
Theorem of Pythagoras
Equations 1
Subtracting Fractions
Solving Quadratic Equations by Graphing
Evaluating Polynomials
Slope
Angles and Degree Measure
   

Evaluating Expressions Involving Fractions

This short note is intended to remind you of principles already described under the heading “Order of Operations,” and to illustrate their application to arithmetic expressions involving fractions.

In evaluating an arithmetic expression, the order in which operations are done is:

(1) bracketed expressions first, starting with the innermost pair of brackets

(2) powers second

(3) multiplications and divisions third

(4) additions and subtractions last Within each priority level, operations are done from left to right.

These rules apply to all arithmetic expressions, including those which involve fractions.

Example:

Simplify

solution:

The expression in brackets gets evaluated first:

thus

Now, the multiplication in the second term gets done, because it has a higher priority than the subtraction:

Finally, we do the subtraction:

Thus, our final answer is

A common error is to start by doing the subtraction:

This is an error, because it carries out the subtraction ahead of the higher priority brackets and multiplication. If carried to completion this will give an incorrect final answer.

Example:

Simplify

solution:

The expression in brackets has the highest priority, so do it first:

So, the original expression becomes

Of the remaining operations, the multiplication has the highest priority, so do it next:

Now, we’re left with subtraction and addition. Both of these operations have the same priority, so we work from left to right. First, subtract 5 / 16 from 7 / 8 :

so that

Since 67 is a prime number, this fraction cannot be simplified further, so the final answer here is

Note that if the addition and subtraction were done in the reverse order (thus violating the priority rules), we would get

and then

which is very different from the correct answer. So, to get the correct answer, the conventional priority rules must be followed very carefully.

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