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Solving Linear Equations
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Evaluating Expressions and Solving Equations
Fraction rules
Factoring Quadratic Trinomials
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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
Percents
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
Algebra
Common Logs
Operations on Signed Numbers
Multiplying Fractions in General
Dividing Polynomials
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
fractions
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
Quadratics
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
Decimals
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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Order of Operations

Look at “4 + 5 × 2”. How do you know which to do first? Should you add 4 plus 5 and then multiply by 2, or multiply first and then add?

Natural Order

There is a “natural” order of operations to mathematics:

1. If there’s any parentheses ( ) then do them first.

2. Then do exponents xn and powers 103 and radicals .

3. Then do multiplication × and division ÷.

4. And then do any addition + and subtraction - from left to right.

It may help to remember them as P.E.M.D.A.S. using their first letters.

Thus in the example 4 + 5 × 2, we would multiply first.

 

Example:

8 - 4 - 1 = ?

Answer:

Since we have more than one subtraction, we do the left one first:

(8 - 4) - 1 = 3

 

Parentheses

What if we wanted to tell someone to do the subtraction in a different order? Use parentheses to change the natural order. Expressions in parentheses are always done first. Note: Singular is parenthesis, and plural is parentheses.

Example:

8 - (4 - 3) = ?

Answer:

The parentheses tell us to do “four minus one” first:

8 - (4 - 3) = 7

What about nested parentheses? Evaluate the innermost parentheses first.

 

Example:

( 8 - (4 + 2))2 = ?

Answer:

( 8 - 6)2 = (2)2 = 4

 

Exponents

Suppose you have an expression with both subtraction and exponents. You should do the exponent first (unless commanded otherwise by parentheses).

Example:

25 - 24 = ?

Answer:

Since there is no parentheses ( ) we do “two to the fourth power” first: 25 - 16 = 9

 

Calculator Warning!

See your calculator’s manual to read about its order of operations. The cheapest ones simply do everything left-to-right as you enter the operations. The better ones save the results of multiplication and then add (or subtract) the products together. Try this on your calculator:

2 ×3 + 4 × 5 = ?

A very basic calculator solves it like this, as you enter it from left-to-right:

2 × 3 + 4 × 5

6 + 4 × 5

10 ×

50

A better calculator will follow the natural order of operations, and give a different answer even though you pressed the same keys in the same order:

2 × 3 + 4 × 5

6 + 4 × 5

6 + 20

26

So be sure to understand how your own calculator handles its order of operations!

By the way, most calculators that follow the natural order (multiplication first) also have parentheses keys so you can change the order, if needed. If you see parentheses on a calculator, that’s a good clue that it was designed to handle the natural order of operations. If in doubt, use the parentheses keys “(“ and “)” to ensure it follows the order you want.

 

More Examples

If a = 4 and b = 3 compute the result:

Example 1:

a + 6 - b = ?

Answer:

Add and subtract from left to right: 4 + 6 - 3 = 7

Example 2:

a / 2 × b = ?

Answer:

Multiply and divide from left to right: 4 / 2 × 3 = 6

Example 3:

b - a / 2 = ?

Answer:

The division comes first, then the subtraction: 3 - 4 / 2 = 1

 

Handling Fractions

One last complication: What about the expression

To figure this out, we must tell you there are always implied parentheses around the numerator and denominator of a fraction. So first do addition and subtraction, and then do the division:

 

What about cascading fractions such as ? This can be either or which give different results. The writer must be clear when putting it on paper! This is another great reason to love parentheses.

Example:

Answer:

Use the implied parentheses around each fraction, then start with the innermost and work your way out.

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