Über die Autorin bzw. den Autor

Mary Jane Sterling is a professor of mathematics at Bradley University in Peoria, Illinois. She has been teaching mathematics for more than 30 years.

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CliffsNotes Algebra I Practice Pack

John Wiley & Sons

Chapter One

Think of this first chapter as a resource or reference for much of what follows. You can come back to review this material if something puzzles you in a later chapter. These underlying principles make up much of every kind of mathematics.

Order of Operations

Mathematics deals in so many symbols. This is a good thing. The use of symbols is efficient; it saves time and writing. It also makes the language of mathematics universal-understood worldwide. Likewise, a universal agreement exists about how operation symbols such as add, subtract, and so on are to be handled in an equation. The order in which things are done makes a difference. Think of putting the cap on a bottle of soda pop and then shaking the bottle. The result is a lot different than if you first shake the bottle and then put the cap on. The same is true here; order makes a difference. Many operations are used in mathematics, and, accompanying them, some rules and conventions need to be followed. These rules or procedures were established so that anyone reading a mathematical statement written by someone else would know exactly what was intended. Mathematicians throughout the world use the same rules.

Basic Order of Operations

When more than one operation is indicated in an algebraic expression, the operations are done in the following order, except when grouping symbols, such as parentheses, interrupt:

First: Powers and roots Second: Multiplication and division Third: Addition and subtraction

If more than one of the same level of operation appears in the expression, do them in order, moving from left to right.

Example Problems

These problems show the answers and solutions.

1. Simplify 50 - [2.sup.2] 6.

answer: 26

By the Order of Operations, calculate the power first, then the multiplication, and then the subtraction.

First do the power, 50 - [2.sup.] 6 = 50 - 4 6.

Then multiply, 50 - 4 6 = 50 - 24.

Finally subtract, 50 - 24 = 26.

2. Simplify [square root of 625] - 2 6 + [7.sup.2].

answer: 62

By the Order of Operations, the root of 625 and the power of 7 are done first.

[square root of 625] - 2 6 + [7.sup.2] = 25 - 2 6 + 49

Next, multiply the 2 and 6, 25 - 2 6 + 49 = 25 - 12 + 49.

Then add and subtract. Since addition and subtraction are on the same level, perform the operations moving from left to right.

25 - 12 + 49 = 13 + 49 = 62

Grouping Symbols

Grouping symbols can "interrupt" the Order of Operations. The most commonly used grouping symbols are parentheses ( ), brackets , and braces { }. Also, fraction lines and radicals (root symbols) act to group expressions above, below, and inside them. The rule is that you perform the operations within the grouping symbols first and then go to the Order of Operations. Grouping symbols more often than not help clarify what is meant in a mathematical statement. Think of them as being like punctuation in a written statement-helping you to understand the meaning.

Example Problems

These problems show the answers and solutions.

1. Simplify 6(14 - 3) + 8.

answer: 74

Perform what's in the parentheses first. 6(14 - 3) + 8 = 6(11) + 8

Then multiply and, finally, add. 6(11) + 8 = 66 + 8 = 74

2. Simplify 3 [square root of 16 - 7] + (5 - 3) 7 - 14/9 - 2.

answer: 21

Three grouping symbols are here: radical, parentheses, and fraction line. Perform the operations within, above, or below them first.

3 [square root of 16 - 7] + (5 - 3) 7 - 14/9 - 2 = 3[square root of 9] + (2) 7 - 14/7

The root has to be calculated next, because powers and roots are on the first level.

3[square root of 9] + (2) 7 - 14/7 = 3 3 + (2) 7 - 14/7

Now, do the two multiplications and the division. 3 3 + (2) 7 - 14/7 = 9 + 14 - 2 Now add and subtract, moving from left to right.

9 + 14 - 2 = 23 - 2 = 21

Work Problems

Use these problems to give yourself additional practice.

1. [4.sup.2] + 3 6 - 2

2. 15 - 3/4 - [square root of 9] + 8

3. [6.sup.2] + 9/9 + 5(8 - [2.sup.2]) - 1

4. 10 + [3.sup.2] - 4 ([square root of 36] - 52)

Worked Solutions

1. 32 First raise 4 to the second power.

[4.sup.2] + 3 6 - 2 = 16 + 3 6 - 2

Next, multiply the 3 and 6.

16 + 3 6 - 2 = 16 + 18 - 2

Last, add and subtract from left to right.

16 + 18 - 2 = 34 - 2 = 32

2. 8 First, subtract the 3 from 15, because they're "grouped."

15 - 3/4 - [square root of 9] + 8 = 12/4 - [square root of 9] + 8

Next, find the square root of 9.

12/4 - [square root of 9] + 8 = 12/4 - 3 + 8

Now, divide 12 by 4 and combine the terms from left to right.

12/4 - 3 + 8 = 3 - 3 + 8 = 0 + 8 = 8

3. 24 First, raise the 6 in the numerator of the fraction to the second power and raise the 2 in the parentheses to the second power.

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Now add the two numbers in the numerator of the fraction and subtract the two numbers in the parentheses.

36 + 9/9 + 5 (8 - 4) - 1 = 45/9 + 5(4) - 1

Next, do the division and multiplication.

45/9 + 5(4) - 1 = 5 + 20 - 1

Last, add and subtract in order from left to right.

5 + 20 - 1 = 25 - 1 = 24

4. 15 First, raise the 3 to the second power and find the square root of 36.

10 + [3.sup.2] - 4 ([square root of 36] - 5) = 10 + 9 - 4 (6 - 5)

Now combine the two numbers in the parentheses and then multiply the result by 4.

10 + 9 - 4(6 - 5) = 10 + 9 - 4(1) = 10 + 9 - 4

Last, add and subtract in order.

10 + 9 - 4 = 19 - 4 = 15

5. 26 First perform the operations in the parentheses and in the denominator of the fraction.

3(6 - 2) + [7.sup.2](5 - 1)/5 + 3 = 3(4) + [7.sup.2](4)/8

Now square the 7.

3(4) + [7.sup.2](4)/8 = 3(4) + 49(4)/8

Now do the multiplications in the numerator. The two terms can't be added until those multiplications are first performed.

3(4) + 49(4)/8 = 12 + 196/8

Add the two numbers in the numerator and then divide the result by 8.

12 + 196/8 = 208/8 = 26

Basic Math Operations

The basic math operations that apply to numbers also apply to variables, which are represented by letters. The main difference between dealing with numbers and variables is that with numbers you can see, directly, what the operation does to them. When dealing with variables, you sometimes don't know what the variable represents, and difficulties could arise depending on whether the variable represents a positive or negative number, a fraction or whole number, an even or odd number, and so on. Following is a discussion of the basic operations and how variables are handled in each situation.

Addition and Subtraction

When you add and subtract terms with the same variable in them, the coefficient (number multiplier) indicates how many of that variable there are. So just...