Video Math Tutor: Basic Math: Lesson 3 - Operations On Numbers

BASIC MATH A Self-Tutorial by

Luis Anthony Ast Professional Mathematics Tutor

LESSON 3: OPERATIONS ON NUMBERS

Copyright © 2005 All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage or retrieval system, without permission in writing of the author. E-mail may be sent [email protected]

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Disclaimer: This Self-Tutorial is meant for students (7th grade through college) who need a basic review of arithmetic on integers (positive and negative whole numbers) and Order of Operations. It is NOT meant for young children who are learning arithmetic for the first time.

|ABSOLUTE VALUE | F The Absolute Value of a number is the distance that number is from zero on the real number line. Math Symbol:

L

…

The absolute value of a number is NEVER negative. It’s either positive or zero. Formal definition of Absolute Value of x:

This definition means that the absolute value of a real number “x” is just itself if the number is zero or positive, and you “change the sign” if the number is negative. The “–x” does not mean “negative x,” but rather the “opposite” of x. This is discussed in more detail a little later on.

EXAMPLE 1: Find

.

SOLUTION: This is how you can “see” what the absolute value of the number 3 is: 1 unit + 1 unit + 1 unit = 3 units away from 0 | →|→|→| ←d||d→ 0 3 1444442444443

The absolute value of 3, denoted by

is equal to the number of units it is

away from zero, since it is three units away, then 2

.

F

The absolute value of –3 is also 3, since –3 is three units away from zero: 1 unit + 1 unit + 1 unit = 3 units away from 0 | ←|←|←| ←d||d→ –3 0 1444442444443

Together:

← 3 units away from 0

3 units away from 0 →

| ←| →| ←d||d||d→ –3

0

3

1444442444443 1444442444443

EXAMPLE 2: What is SOLUTION:

?

because 4 is four units away from zero. ←d|||d→ 0 4 1444442444443

←

EXAMPLE 3: What is SOLUTION:

four units away

→

?

since 0 is… well… ZERO units away from itself,

it is the ONLY absolute value that is equal to zero. All others are positive. ← d → 0

3

(

Absolute value on most graphing calculators look

like “ ,” you place the value within the parentheses. There are a couple of ways of finding the absolute value of a number using the calculator. Let’s find : What to do: Find :

On the Calculator Screen: Y First Way:

Y First Way: M y
Y Second Way:

Y Second Way: M B

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These Notes are the Copyrighted Property of Luis Anthony Ast © 2005. All Rights Reserved.

4

– OPPOSITE NUMBERS F Opposite Numbers are the same distance from zero on the real number line, but they are on opposite sides on the number line. Math Symbol: – or –( )

EXAMPLE 4: –3 is the opposite of 3 since they are both three units away from zero, but are in opposite directions. ←d|||||d→ –3 0 3 % Opposite Numbers &

L

Opposite numbers have the same absolute value, so…

…

EXAMPLE 5: Y What is the opposite of 2? Answer: –2 Y What is the opposite of –5? Answer: 5 Y What is the opposite of π? Answer: –π Y What is the opposite of 0? Answer: 0 A negative sign in front of a number, variable, or grouping symbol is used to symbolize “the opposite.” So… –x is the opposite of x – (–8) is the opposite of –8, which is 8

5

(

the negation key

is used to find the opposite of

a number of an expression; so it is used to represent both a negative number and the negation (the opposite) of a number. What to do: Find the opposite of –2 M

M

= F

On the Calculator Screen:

Or…

Be sure you use the negation key and NOT the subtraction key when you want to find the opposite of some expression. Opposites are also called Additive Inverses (more on this in Lesson #4).

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ARITHMETIC OF INTEGERS A+D+D+I+T+I+O+N F Addition is the operation of combining numbers to provide an equivalent single value. The numbers being added are called the summands and the result is called the sum. Sum m and + Sum m and = Sum

Math Symbol: +

Addition of numbers can be “visualized” on the number line. Start at the origin , then move to the right, if the number is positive, and left if the number is negative. 6

EXAMPLE 6: Add 2 + 3 on a number line. Step 1: Go 2 units to the right, since 2 is positive: 2 units → ←||||||||→ 0 Step 2: Go 3 units to the right, since 3 is positive: 3 units → ←||||||||→ 0

2

Since we stopped at 5, that is our sum, so 2 + 3 = 5. Let’s throw in a negative number now.

EXAMPLE 7: Add –3 + 4 on a number line. Step 1: Go 3 units to the left, since –3 is negative: ← 3 units ←||||||||→ 0 Step 2: From –3, go 4 units to the right, since 4 is positive: 4 units → ←||||||||→ –3

0

Therefore, –3 + 4 = 1

7

The previous example may also be done on a single number line:

←||||||||→ –3 0 1 –3 + 4 = 1 It is cumbersome to always use the number line for addition, so we generally just add numbers mentally, use our fingers and toes, or simply use a calculator. We use the following rules to help us with addition:

RULES OF ADDITION Rule 1 : “positive” + “positive” = “positive” Add the numbers, the sign of the sum is also positive. Rule 2 : “negative” + “negative” = “negative” Add the numbers, the sign of the sum is also negative. Rules 1 and 2 mean: if you have “like” and place the sign you see for the final sum.

signs, add up the numbers,

Rule 3 : “positive” + “negative” or “negative” + “positive” This rule is a little more complicated… What you do is temporarily ignore the signs you see, subtract the “smaller” number from the “larger,” then include the sign of the number whose absolute value was bigger (whichever number is bigger without looking at the plus/minus signs, is the sign you will use for the final result.) Don’t worry, we will do a few examples of this to make it easier to learn. Rule 4 : “any number” + “its opposite” = 0 This is called the Additive Inverse property of addition. The opposite is the additive inverse of a number. Rule 5 : 0 + “a number” or “a number” + 0 = “the number” This is called the Additive Identity property of addition. Zero is the additive identity. 8

EXAMPLE 8: Here are some mini-examples. Try doing them “by hand” first, then use your calculator. One of the keys to successfully using your calculator is to and PRACTICE. Yes, these are simple problems, but my recommendation is to always try to do problems by hand first (if possible), then use a calculator. Why? Well, if you only use your calculator for the “tough problems,” you may not have the experience to do the problem. Students may “blank” when trying to find the right keystrokes to do a problem. The more you practice, the better you get. PRACTICE, PRACTICE,

(

What to do: Example of Rule 1 : Find 8 + 5 M

On the Calculator Screen:

Example of Rule 2 : Find the sum of –8 and –10 M Example of Rule 3 : Add –12 and 6 M Another example of Rule 3 : What is five plus negative four? M Example of Rule 4 : Find the sum of 9 and its opposite. M Example of Rule 5 : What is 0 + 7? M

9

L

…

When using a calculator, you don’t need to enclose the negative number within parentheses, but you may do so, for clarity, when writing it out on paper. This way, you don’t confuse the negative sign with the subtraction sign.

Some of our previous problems could be written as follows: 5 + –4 can be written as: 5 + (–4) –8 + –10 is the same as: (–8) + (–10) And 9 + –9 is equal to 9 + (–9)

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SUB–TRAC–TION F Subtraction is the inverse

operation of addition. With addition, you “add” numbers together, while with subtraction, you “take away” numbers. Math Symbol: – The number that is “taken away” from the original number is called the Subtrahend, the original number is called the Minuend, and the result of a subtraction is called the Difference.

Minuend – Subtrahend = Difference

RULE FOR SUBTRACTION To subtract one number from another, substitute the subtrahend by its opposite, then ADD the numbers together. By doing this, all subtractions become additions. 10

EXAMPLE 9: Perform the following operations: Y What is 10 – 3? Solution: 10 – 3 = 10 + (–3) = 7 Y Subtract –5 from 8. Solution: 8 – (–5) = 8 + (5) = 8 + 5 = 13

( subtraction key: the negation key:

To subtract using a calculator, use the to perform the operation. Do not mix up this key with . If you do, this is the error message you will see:

If you see this, press to go to the error. The screen changes back, but the cursor will blink over the error. Just correct it, and press . This should solve the problem. Let’s do a simple subtraction: What to do: Find: –11 – (–6) M

On the Calculator Screen:

Note: Look carefully at the minus and negative signs on the calculator display. They are slightly different. The negative sign is smaller and slightly higher on the calculator display than the minus sign. In these Notes, I follow standard math notation and usually make them the same (but I might change them a little, like using bold type for emphasis). 11

HOT TIP!

When writing negative and minus signs by hand, make them “longer” than you have in the past. This way, you are clear that a number is negative or a subtraction is being performed. I have found over the years that students that write very short signs can confuse them with decimal points, the “=” sign, or not even know they are there. Example: Write

instead of

, which looks like:

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|DISTANCE – BETWEEN | |TWO – NUMBERS | F What if we would like to figure out how far one number is away from another? How can we do this? For example, I want to find the distance between –3 and 4. This is easy to find using a number line: ←||||||||→ –3 0 4 Just count the number of units between –3 and 4: ←||||||||→ –3 0 4 7 Units You can count starting at –3 or from 4. It doesn’t matter. At the beginning of this lesson, we learned that absolute value is used to find the distance a number is away from zero. Now we can combine this with our knowledge of subtractions, to get a special formula that will help us find this distance. The distance between –3 and 4 is =

12

In general: The distance between numbers a and b on a number line is:

F

=

Do not mix up this distance formula for the one that is used to measure the distance between two points in a plane: This formula is discussed in another lesson.

GGGGGGGGGGGGGGGGGG

MUL

TI

PLI

CA

TION

F Multiplication is a short-hand notation for repeated addition or subtraction. Math Symbol:

For example, 2 +2 +2 +2 +2 = 10 can be re-written as a multiplication: 5 × 2 = 10 The “×” symbol is fine to use when you are just multiplying numbers together, but when algebra is involved, it can be confused with the variable “x.” There are several ways to represent multiplication. Let “a” and “b” represent arbitrary numbers. “a” times “b” can be represented by any of the following: Notation: a×b a⋅b

Comments: Not used very much in algebra [See above]. A “raised dot.” Not usually used with numbers, since it may be confused with a decimal point. OK to use with variables. 13

(a)(b) [a][b] {a}{b} ab (a)b a(b)

When there is NO symbol between numbers and variables, or just variables, then this is a way of showing “implied” multiplication. Here, expressions are placed within a set of grouping symbols, but there is nothing between groups of symbols. This is a preferred notation. Also implied multiplication. The variables are just next to each other. This is the most typical way of showing this operation with variables. Not used with numbers. Implied multiplication. Not used very often. Implied multiplication. This is frown upon for multiplication of variables, since it can be confused with another math notation called “function notation.” It’s OK to use with numbers, or numbers (in place of a) with variables (in place of b).

The numbers/variables being multiplied together are called Factors, and the result of a multiplication is called the Product.

(Factor) × (Factor) = Product

RULES OF MULTIPLICATION Rule 1 : “positive” × “positive” = “positive” “negative” × “negative” = “positive” The product of numbers with like signs is positive. Rule 2 : “positive” × “negative” = “negative” “negative” × “positive” = “negative”

or

The product of numbers with different signs is negative. Rule 3 : 0 × “any number” = 0 “any number” × 0 = 0

or

This is called the “Zero-Factor” property of multiplication. 14

These Rules can be abbreviated as:

=

(+)(+) = +

(–)(–) = +

(+)(–) = –

(–)(+) = –

(0)(#) = 0

(#)(0) = 0 #

Be careful and NOT mix up the multiplication and addition rules. For example: (–5) + (–2) = –7, but (–5)(–2) = 10 0 + 5 = 5, but 0 × 5 = 0

A number “x” being multiplied by –1 can be represented as follows (excluding other notations): (x)(–1) = (–1)(x) = –1(x) = (–1)x = –1x = –(x) = –x Note: Notice the last two on right hand side. A negative sign to the left of a parenthesis (or any grouping symbol) may be seen as multiplication by –1 or as finding the opposite of a number.

(

The

key is used to represent multiplication. To

differentiate it from the letter , the calculator displays the multiplication using an asterisk: . To demonstrate… What to do: What is –13 × 27? M

On the Calculator Screen:

The Texas Instruments® graphing calculators understand “implied” multiplication, so all of the following are the same as –13 × 27: –13 * 27 –13(27) (–13)27 (–13)(27) (–13)*(27)

15

=

A “raised dot” is OK for multiplication “on paper,” but NOT on a calculator. The key is to enter a decimal point ONLY.

The same can be said for other grouping symbols: { } and [ ]. These have different meanings on a calculator. So if you need to enter the following in your calculator: 3[ (4 + 8) – 2 ] Use an additional set of parentheses in place of the square brackets: 3( (4 + 8) – 2 )

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E

X

PONENTS

F Just as multiplication is an abbreviated form of repeated addition, Exponents are used to abbreviate repeated multiplication. Math Symbol:

Using the number 3 as an example, this is what is meant by repeated multiplication: 3×3×3×3×3= = 243 5 factors of 3 is read as: “three to the fifth power” or simply: “three to the fifth.” Using a little algebra now: x ⋅ x ⋅ x ⋅ …⋅ x = n factors of x is read as: “x to the nth power” or simply: “x to the n.” 16

F

The expression: 3 × 3 × 3 × 3 × 3 is said to be in

Expanded Form while is in Exponential Form, Exponential Notation, or Exponent Notation. Some terminology:

x

n

This is the Power or Exponent

↑

This is the Base is read: “x raised to the first power” or “x to the first.” Note: If a number or variable is raised to the first power, we can write it without the “1,” that is:

.

is read: “x squared,” “x raised to the second power” or “x to the second.”

=Don’t say: “x two” or “x to the two.”

is read: “x cubed,” “x raised to the third power” or “x to the third.”

=Don’t say: “x three” or “x to the three.” Similar warnings for the rest below. is read: “x raised to the fouth power” or “x to the fourth.” is read: “x raised to the fifth power” or “x to the fifth.” Oh…

(

is read: “x raised to the yth power” or just “x to the y.” Use the power key:

to input an exponent. If

you just want to square a value, use the key. There is an option to input a cube, but I find students don’t ever use it, so don’t worry about it. Note: On some calculators, the “power” or “exponent” keys may look like: xy, yx , or ab . Also, looking at your calculator, there are other keys/functions that have powers in them, like: but we will discuss these in other Lessons. 17

,

,

(among others),

Another Note: When I am describing keystrokes to students, I say “power” when I refer to this key: . For example, I would say is: “three power four.” It’s a personal choice. You can still just say: “three to the fourth power.” Let’s do some calculator drills: What to do: What is M

On the Calculator Screen:

?

Or: M

Of course, you could also just type: M

What is negative three raised to the fourth power?

=Since we are raising a negative number to a power, it MUST be enclosed within parentheses, otherwise, you may get the wrong answer (depending on the power). So we want:

, NOT:

M This is the incorrect way: M is really the opposite of

.

We get the wrong answer.

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DIVI ÷ SION

F Division is the inverse or reverse operation of multiplication. It tells us how many times one number is contained within another number. Math Symbol: ÷

The number being divided is called the Dividend. The number doing the dividing is called the Divisor. The result of a division is called the Quotient.

Dividend ÷ Divisor = Quotient To divide a number “x” by a number “y,” we can use any of the following notations: Notation: x÷y x/y or

or y x

Comments: Not used very much in algebra. OK to use. Appears in operations dealing with fractions. Great for numbers. OK to use, but may be a little vague when dealing with complicated expressions. Used typically to write fractions in a more compact form, such as in an exponent or matrix. This is the best way to express division in algebra, especially when the items get more complicated. The top part is called the Numerator and the bottom part is called the Denominator. Try to use this notation when possible. These are used when performing “long division” and are the same as:

If a number does not exactly divide into another number, then there is a Remainder.

F

Dividend = (Divisor) 19

(Quotient) + Remainder

EXAMPLE 10: Using some numbers: 20 ÷3 or

is:

RULES OF DIVISION Rule 1 : “positive” ÷ “positive” = “positive” “negative” ÷ “negative” = “positive” The quotient of numbers with like signs is positive. Rule 2 : “positive” ÷ “negative” = “negative” “negative” ÷ “positive” = “negative”

or

The quotient of numbers with different signs is negative. Rule 3 : 0 ÷ “any non-zero number” = 0 Rule 4 : “any non-zero number” × 0 = undefined.

F

We won’t consider the

case. This is discussed in

calculus. These Rules can be abbreviated as:

=

Dividing by zero is NOT allowed! Zero cannot be in the denominator of a fraction. Don’t go there! Do not pass Go. Do not collect $200. Oops! I got a little carried away there. Sorry. L 20

F RECIPROCALS F Y The Reciprocal of a number x is Y The reciprocal of a fraction

.

is N

O

Y A number multiplied by its reciprocal is always equal to 1. Here are a couple of examples: and This is called the Multiplicative Inverse property. Y The reciprocal itself is also called the multiplicative inverse. Y Zero is the only number that does not have a reciprocal. Y With reciprocals, all divisions can now be re-written as multiplications: a ÷ b is the same as:

(

The

,

or

key is used to perform divisions. It is

displayed on the calculator screen as a “forward slant:” /. The reciprocal of a number can be found by using the reciprocal key: . In practice, this key is very rarely used. Most students just use: then the number.

L HOT TIP!

…

One of the places many students make mistakes using the calculator is in the area of divisions. Be extra careful any time you need to perform this operation or doing fractions on the calculator.

When in doubt, just enclose the entire numerator within an extra set of parentheses and do likewise with the entire denominator.

EXAMPLE 11: To evaluate

using the calculator, “visualize” this as: 21

.

The expression:

should be “seen” as:

(

The last two expressions would be entered as:

What to do: What is

On the Calculator Screen: ?

M

What is

?

M

F NEGATIVES WITH DIVISION F Y The following are all equal: =

=

=

Y This equation is always true:

F OTHER EQUATIONS WITH DIVISION F Equation:

Comments: Any number, over itself, is one. A number, divided by 1, is just itself. 22

ORDER + OF × OPERATIONS Now that we have all these arithmetic operations, what happens when we combine them? How do we evaluate something like: 2 + 3 × 4 ? Do we add the 2 and the 3, then multiply by 4, or do we multiply the 3 and the 4 first, then add 2 to this? Before we can answer this, we need to talk about the order of operations. We will start this discussion with grouping symbols.

F Grouping Symbols are used to show certain mathematical operations should be done before others in an expression. Here is a list of the most common symbols used in grouping: Symbols:

Comments: These are the most commonly used grouping symbols. Example: 1 – (2 + 3)

( or Square Brackets or Curly Brackets

Only use parentheses for grouping. Used to enclose items that already have parentheses. Example: 4[1 – (2 + 3)]

(

Used for matrices. Used to group items that have square brackets. Example: 6 + {5 – 4[1 – (2 + 3)]}

(

Used to enclose items in lists. Used to group items that have braces.

or Vinculum Example: First, evaluate everything in the numerator. Next, evaluate everything in the denominator. Finally, divide the numerator by the denominator. A detailed example of this will be shown later. Absolute Do everything within the bars first, then take its absolute Value value. Bars A detailed example of this will be shown later. Do everything “inside” the radical first, then take the root. . Examples of this are in the Algebra Lesson: “Radicals.” the radical sign is really just “ √ ”

F

The vinculum

is used to “extend” the sign: “ 23

”

There are other grouping symbols/operations, but we won’t encounter them until other lessons.

F ALTERNATE GROUPING NOTATION F Say we are given something that looks like this:

Which is rather complicated looking. Instead of using braces and bars, the parentheses and brackets can alternate, so the previous problem would look like:

OK, it’s still a mess, but it is a little “easier on the eyes.”

L

…

We don’t have to use parentheses. It’s perfectly acceptable to write any of the following for grouping: =

=

=

But it’s usually best to stick with parentheses.

F ORDER OF OPERATION RULES F Y Evaluate expressions within grouping symbols. So, if you have, say, 1 + (8 – 3), you would do 8 – 3 first. If you have items grouped within another set of grouping symbols, evaluate first the “inner” set of grouping symbols. An example of this would be: 5 + 4[3 – (1 + 2)]. You would evaluate (1 + 2) first, since it is the innermost set of grouping symbols, then take care of everything inside of the brackets. Y Perform all exponential expressions before other arithmetic operations. Given:

, you need to square the 4 first, NOT add the 3 with the 4.

24

Y Next, perform all multiplications and/or divisions , but evaluate the expression in order from left-to-right. To evaluate 3 × 2 + 4 ÷ 2, for example, multiply 3 × 2 first, since it is to the left of 4 ÷ 2. This gives us: 6 + 4 ÷ 2. Now, divide the 4 by 2. Our expression will now look like this now: 6 + 2. Finally, do the addition and the result is 8. Y The last rule is to do all additions and/or subtractions but evaluate the expression in order from left-toright. Let’s evaluate 8 – 3 + 5 –1. Going from left-to-right, do 8 – 3 first. This gives us 5: 5 + 5 – 1. Now add the fives to get 10: 10 – 1. Finally, do the subtraction to get 9 as the end result.

ORDER OF OPERATIONS 1 First, evaluate the expression within the innermost set of grouping symbols. 2 Next, evaluate expressions that have exponents. 3 Then, perform all multiplications and divisions, going from left-to-right in the expression. 4 Finally, do all additions and subtractions, again, going from left-to-right in the expression.

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An acronym to help you remember the above is:

HOT TIP!

PEMDAS P: Parentheses E: Exponents M: Multiplication D: Division A: Addition S: Subtraction

PEMDAS can be memorized easily if you remember the following mnemonic device:

“Please Excuse My Dear Aunt Sally” F ORDER OF PRECEDENCE F

Y The Order of Precedence, is the order of importance in performing operations. The “higher” the precedence, the more “important” it is; therefore, needs to be done before items of a “lower” precedence. The order is the same as the order of operations: 1 Grouping symbols 2 Exponents 3 Multiplications / Divisions 4 Additions / Subtractions There are additional operations with other orders of precedence, but these will do for now. Note: Most graphing calculators and computer programs follow the Order of Precedence mentioned here. Smaller, non-graphing calculators may evaluate expressions in a different way, so be careful if using them. Consult your calculator’s Operating Manual for more information Now that we have all of these rules, let’s do several detailed examples. First, “by hand,” showing all steps, then we will verify the answer by evaluating the expressions directly using the calculator. 26

2

EXAMPLE 12: Simplify: 4 + 3 × 2 – 10

SOLUTION: Y There are no grouping symbols, so start with the exponent: 2

4 + 3 × 2 – 10 = 4 + 9 × 2 – 10 Y Next, do the multiplication:

4 + 9 x 2 – 10 = 4 + 18 – 10

Y Now, since we only have additions and subtractions, evaluate the expression going from left-to-right: 4 + 18 – 10 = 22 – 10 Y Finally, do the subtraction:

( What to do:

22 – 10 = 12

On the Calculator Screen: 2

Simplify: 4 + 3 × 2 – 10 M

EXAMPLE 13: Perform the indicated operations: 2

–5 + 2[3 – 4(7 – 6) + 2 ]

SOLUTION: Y Evaluate what is in the innermost parentheses first: 2

–5 + 2[3 – 4(7 – 6) + 2 ]

= –5 + 2[3 – 4(1) + 22] 27

Y Do the Exponent next. The parentheses around the 1 are used for “implied” multiplication, so the exponent is evaluated before the multiplication: 2

–5 + 2[3 – 4(1) + 2 ] = –5 + 2[3 – 4(1) + 4] Y Perform the multiplication within the brackets: –5 + 2[3 – 4(1) + 4] = –5 + 2[3 – 4 + 4] Y Evaluate the subtraction and addition inside the brackets (going from left-to-right): –5 + 2[3 – 4 + 4] = –5 + 2[–1 + 4] = –5 + 2[3] Y The brackets are now used as “implied” multiplication, so perform that next: = –5 + 2[3] = –5 + 6 Y Finally, add the numbers together:

= –5 + 6 =1 This seems like we are doing a huge number of steps, but I am doing this step-by-step, and I re-display certain steps for clarity. Here’s how this problem would look like if it were done “by hand” for a test question: 2 –5 + 2[3 – 4(7 – 6) + 2 ]

= –5 + 2[3 – 4(1) + 22] = –5 + 2[3 – 4(1) + 4] = –5 + 2[3 – 4 + 4] = –5 + 2[–1 + 4] = –5 + 2[3] = –5 + 6 28

=1 If your instructor allows you to “combine” steps, the above problem can be done in even fewer steps, but I only recommend you do this after a great deal of practice . 2

–5 + 2[3 – 4(7 – 6) + 2 ] = –5 + 2[3 – 4(1) + 4] = –5 + 2[3 – 4 + 4] = –5 + 2[3] = –5 + 6 =1

(

What to do: Perform the indicated operations: 2 –5 + 2[3 – 4(7 – 6) + 2 ] M

On the Calculator Screen:

Remember to use parentheses only, not brackets, when entering the keystrokes into your calculator.

EXAMPLE 14: Simplify the expression using the order of operation rules: 2 – |5 – 2 × 8| 3

SOLUTION: Y The absolute value bars serve as grouping symbols, so evaluate what is inside them first: 2 – |5 – 2 × 8| 3

Y Multiplication has higher precedence: 2 – |5 – 2 x 8| 3

= 2 – |5 – 16| 3

29

Y Now do the subtraction within the absolute value: 2 – |5 – 16| 3

= 2 – |–11| 3

Y Next, perform the absolute value: |–11| = – 11 2 – |–11| 3

3

= 2 – 11 Y Then, evaluate the exponent: 3

2 – 11 = 8 – 11 Y Finally, do the subtraction: 8 – 11 = –3

( What to do:

On the Calculator Screen:

Simplify: 2 – |5 – 2 × 8| M y
EXAMPLE 15: Simplify:

Basic Strategy:

The fraction bar serves as a grouping symbol. So… 1 Evaluate the entire numerator. 2 Evaluate the entire denominator. 3 Divide the numerator by the denominator.

SOLUTION: Y Starting with the numerator, evaluate the exponents: 30

=

Y Evaluate all subtractions and additions in the numerator, going from left-to-right: =

=

Y In the denominator, evaluate the subtraction within the parentheses first: =

Y Evaluate the exponent next: =

Y Now, do the subtraction: = Y Finally, divide the numerator by the denominator (leaving it in fractional form): = You could have also just written 0.5 as the final answer. An alternate way to evaluate the original problem is to simplify the numerator and the denominator at the same time. Starting with:

31

Y Exponentiate the items in the numerator and do the subtraction in the denominator:

Y Do the subtraction in the numerator, and square the 2 in the denominator:

Y Perform the addition on top, and the subtraction below:

Y Finally, divide the top by the bottom (leaving it in fractional form): =

(

Be sure to enclose the entire numerator and

denominator within an extra set of parentheses. In other words, change:

What to do:

into:

On the Calculator Screen:

Simplify: M

Convert to a fraction: 32

LESSON 3 QUIZ When doing these problems, try to also do them using your calculator, (if possible) to get more practice using it.

1 Write the following expressions without absolute value bars, simplifying also, if possible: Y Y Y Y Y

2 Rewrite the following without absolute values, leaving the answer in EXACT form: Y Y Y

3 Find the opposites of the following: Y 5. The opposite is: _____ Y –8. The opposite is: _____ 33

Y

. The opposite is: _____ . The opposite is: _____

Y

Y –x. The opposite is: _____

4 Perform the following operations: Y 8 + (–2) = Y –8 + 2 = Y –8 + (–2) = Y Subtract –2 from –8: Y –8 – 2 = Y8×2= Y –8(2) = Y (–8)(– 2) = Y8÷2= Y Y

= =

Y Find the square of negative five = Y

=

Y

=

34

5 What is the distance between –5 and 7? 6 What is the distance between x and y? 7 Which of the following is considered

as the

BEST way to input “two times three” in a calculator? (2)(3) (2) (3)

2(3) 2 (3)

(2)3 (2) 3

8 What is the expanded form of

2 3

?

9 Write the Exponential Notation for 7 × 7 × 7: bl What is the base of

?

bm What is the exponent of

?

bn Which operation is performed first for:

?

bo Which of the following are not typical grouping symbols used in math expressions?

bp What is PEMDAS? bq True or False. Subtraction has a higher order of precedence than Division. ________

35

br Simplify. Show all steps. Write work on a separate sheet of paper. Y –3 × 2 + 8 = Y (2 + 3) × 5 = Y (6 – 2)(8 + 1) = Y Y

Y

Y

Y Y

=

Y

Y

ANSWERS ON NEXT PAGE… 36

ANSWERS 1 Write the following expressions without absolute value bars, simplifying also, if possible: Y Y

8 7.2

Y

6

Y

10 – 16 = –6 –5

Y

2 Rewrite the following without absolute values, but leaving the answer in EXACT form: Y

f–2

Explanation: since 2 – f is negative, you want to get a positive value, since absolute values are always positive, so the way around this is to just “switch” the places of the numbers. Y

f – A7

Explanation: since is already positive (try it on a calculator), you just need to remove the absolute value bars. Y

3 – A7

Explanation: Similar to first Y in Problem 2. These types of questions are very typical on tests.

37

3 Find the opposites of the following: Y 5. The opposite is: –5 Y –8. The opposite is: 8 Y Y

. The opposite is: – A3 . The opposite is: –6

Y –x. The opposite is: x

4 Perform the following operations: Y 8 + (–2) = 6 Y –8 + 2 = –6 Y –8 + (–2) = –10 Y Subtract –2 from –8: This is written as: –8 – (–2) = –8 + 2 = –6 Y –8 – 2 = –8 + (–2) = –10 Y 8 × 2 = 16 Y –8(2) = –16 Y (–8)(– 2) = 16 Y8÷2=4 Y Y

= –4 = 4 × 4 × 4 = 64 (or just use calculator to get this value). 38

Y Find the square of negative five =

25

Note: The answer is NOT –25, since we want:

.

A typical error is to write what was asked for as:

Y

=1

Y

= 301

5 What is the distance between –5 and 7? Using the distance formula, we get:

F

12

We could have also written:

12

6 What is the distance between x and y? Using the distance formula again, we get: y – x

F

We could have also written: x – y

7 Which of the following is considered as the BEST way to input “two times three” in a calculator? (2)(3) (2) (3)

F

2(3) 2 (3)

(2)3 (2) 3

D2 3

All of the above are acceptable. I want you to use

the one that would involve the fewest keystrokes; however, if you need to add more keystrokes so that the expression seems clearer to you, then go ahead and add more.

8 What is the expanded form of

?

Answer: 6 × 6 × 6 × 6 39

9 Write the Exponential Notation for 7 × 7 × 7: Answer: 73

bl What is the base of

? Answer: The base is 5. The base is NOT –5. By “order of precedence,” the square is just with the 5, and does not include the negative sign. If we wanted the base to be –5, then the expression should have been written:

.

bm What is the exponent of

? Answer: The exponent is 5.

bn Which operation is performed first for:

? Answer: The Exponent or Square the Four. Exponents are done before the other operations.

bo Which of the following are not typical grouping symbols used in math expressions?

D

bp What is PEMDAS? Answer: It is an acronym to help you remember the order of operations of real numbers. The letters stand for: P: Parentheses E: Exponents M: Multiplication D: Division A: Addition S: Subtraction

bq True or False. Subtraction has a higher order of precedence than Division. Answer: FALSE. Subtraction (along with addition) has the lowest order of precedence presented in this lesson. 40

br Simplify. Steps are shown below each problem. Y –3 × 2 + 8 = 2 Answer:

–3 x 2 + 8 = –6 + 8 =2

Y (2 + 3) × 5 = 25 Answer:

(2 + 3) × 5 = (5) × 5 = 25

Y (6 – 2)(8 + 1) = 36 Answer: (6 – 2)(8 + 1) = (4)(9) = 36 Y Answer:

4 2(1 – 3 + 22) 2

= 2(1 – 3 + 2 ) = 2(1 – 3 + 4) = 2(1 – 3 + 4) = 2(–2 + 4) = 2(–2 + 4) = 2(2) =4 Y Answer:

12 10 – [2 + (4 – 23)] 3

= 10 – [2 + (4 – 2 )] = 10 – [2 + (4 – 8)] = 10 – [2 + (4 – 8)] = 10 – [2 + (–4)] = 10 – [2 + (–4)] 41

= 10 – [–2] = 10 + 2 = 12 Y

64

Answer: = = 64 6

Y Answer:

12 + 2[–4 + (2 – 3)2] = 12 + 2[–4 + (–1)2] 2

= 12 + 2[–4 + (–1) ] = 12 + 2[–4 + 1] = 12 + 2[–4 + 1] = 12 + 2[–3] = 12 + 2[–3] = 12 + (–6) =6 Y Answer:

97 – | –3| + (8 + 2)2 = –(3) + (10)2 2

= –(3) + (10) = –(3) + 100 = –3 + 100 = 97

42

= –20

Y Answer: = = = = = = = = = = = –20 Y

4

Answer:

= = =

43

= =4

–5

Y

Answer:

=

=

=

=

= = = –11 – (–6) = –11 + 6 = –5

END OF LESSON 3 44