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# Order of Operations

## The order of operations outlines the order by which certain operatives should take place in a mathematical expression for it to remain true

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Algebraic Properties

Abigail buys 2 shirts at $12 each. She also buys a pair of jeans that were originally$50 but are discounted by 15%. If her total purchase is over $60, she will get an additional$10 off. How much was her bill?

### Algebra Properties

The properties of algebra enable us to solve mathematical equations. Notice that these properties hold for addition and multiplication.

Property Example
Commutative \begin{align*}a+b=b+a,ab=ba\end{align*}
Associative \begin{align*}a+(b+c)=(a+b)+c,a(bc)=(ab)c\end{align*}
Identity \begin{align*}a+0=a,a \cdot 1=a\end{align*}
Inverse \begin{align*}a+(-a)=0,a \cdot \frac{1}{a}=1\end{align*}
Distributive \begin{align*}a(b+c)=ab+ac\end{align*}

From the Identity Property, we can say that 0 is the additive identity and 1 is the multiplicative identity. Similarly, from the Inverse Property, \begin{align*}-a\end{align*} is the additive inverse of \begin{align*}a\end{align*} and \begin{align*}\frac{1}{a}\end{align*} is the multiplicative inverse of \begin{align*}a\end{align*} because they each equal the associated identity.

Let's identify the property used in the equations below.

1. \begin{align*}2(4x-3)=8x-6\end{align*}

This is an example of the Distributive Property.

1. \begin{align*}5 \cdot \frac{1}{5}=1\end{align*}

This is an example of the Inverse Property.

1. \begin{align*}6 \cdot (7 \cdot 8)=(6 \cdot 7) \cdot 8\end{align*}

This is an example of the Associative Property.

Going along with the properties of algebra is the Order of Operations. The Order of Operations is a set of guidelines that allow mathematicians to perform problems in the exact same way. The order is as follows:

Parenthesis: Do anything in parenthesis first.

Exponents: Next, all powers (exponents) need to be evaluated.

Multiplication/Division: Multiplication and division are done at the same time, from left to right.

Addition/Subtraction: Addition and subtraction are also done together, from left to right.

Next, let's solve the following problems using PEMDAS.

1. Simplify \begin{align*}2^2+6 \cdot 3-(5-1)\end{align*}.

First: Parenthesis \begin{align*}\rightarrow 2^2+6 \cdot 3-4\end{align*}

Next: Exponents \begin{align*}\rightarrow 4+6 \cdot 3-4\end{align*}

Then: Multiplication \begin{align*}\rightarrow 4+18-4\end{align*}

Finally: Add/Subtract \begin{align*}\rightarrow 18\end{align*}

1. Simplify \begin{align*}\frac{9-4 \div 2+13}{2^2 \cdot 3-7}\end{align*}.

Think of everything in the numerator as being in its own set of parenthesis, and everything in the denominator as being in another set. The problem can be rewritten as \begin{align*}(9-4 \div 2 +13) \div (2^2 \cdot 3-7)\end{align*}. When there are multiple operations in a set of parenthesis, use the Order of Operations within each set.

\begin{align*}&(9-4 \div 2 +13) \div (2^2 \cdot 3-7)\\ &(9-2+13) \div (4 \cdot 3-7)\\ &(7+13) \div (12-7)\\ & 20 \div 5\\ &4\end{align*}

Parenthesis can also be written within another set of parenthesis. This is called embedding parenthesis. When embedded parenthesis are in a problem, you may see brackets, [ ], in addition to parenthesis.

### Examples

#### Example 1

Earlier, you were asked to find the total cost of Abigail's bill.

Without knowing the Order of Operations, Abigail could get very confused and not have the right amount of money to pay for her items. So, the two shirts are 24 \begin{align*}(2 \cdot \12)\end{align*} and only the jeans are discounted. They are \begin{align*}50-0.15\cdot 50 = 42.50\end{align*}. Adding these two items together, we get that her total bill is24 + $42.50 =$66.50. So, because her purchase is over $60, she gets an additional$10 off, making her final bill \$56.50.

#### Example 2

What property is being used?

\begin{align*}5(c - 9) = 5c - 45\end{align*}

5 is being distributed to each term inside the parenthesis, therefore the Distributive Property is being used.

#### Example 3

What property is being used?

\begin{align*}6 \cdot 7=7 \cdot 6\end{align*}

Here, order does not matter when multiplying 6 and 7. This is an example of the Commutative Property.

#### Example 4

Use the Order of Operations to simplify \begin{align*}8+[4^2-6 \div (5+1)]\end{align*}.

This is an example of embedded parenthesis, as discussed above. Start by simplifying the parenthesis that are inside the brackets. Then, simplify what is inside the brackets according to the Order of Operations.

\begin{align*}&8+[4^2-6 \div (5+1)]\\ &8+[4^2-6 \div 6]\\ &8+[16-6\div 6]\\ &8+[16-1]\\ &8+15\\ &23\end{align*}

### Review

Determine which algebraic property is being used below.

1. \begin{align*}8 + 5 = 5 + 8\end{align*}
2. \begin{align*}7(x - 2) = 7x - 14\end{align*}
3. \begin{align*}\frac{2}{3} \cdot \frac{3}{2}=1\end{align*}
4. \begin{align*}4 \cdot (5 \cdot 2)=(4 \cdot 5) \cdot 2\end{align*}
5. \begin{align*}-\frac{1}{4}+0=-\frac{1}{4}\end{align*}
6. \begin{align*}-6+6=0\end{align*}
7. What is the additive inverse of 1?
8. What is the multiplicative inverse of \begin{align*}-\frac{1}{5}\end{align*}?
9. Simplify \begin{align*}6(4-9+5)\end{align*}using:
1. the Distributive Property
2. the Order of Operations
3. Do you get the same answer? Why do you think that is?

Simplify the following expressions using the Order of Operations.

1. \begin{align*}12 \div 4+3^3 \cdot 2-10\end{align*}
2. \begin{align*}8 \div 4+(15 \div 3-2^2) \cdot 6\end{align*}
3. \begin{align*}\frac{10-4 \div 2}{7 \cdot 2+2}\end{align*}
4. \begin{align*}\frac{1+20-16 \div 4^2}{(5-3)^2+12\div 2}\end{align*}
5. \begin{align*}[3+(4+7 \cdot 3)\div 5]^2-47\end{align*}
6. \begin{align*}\frac{7 \cdot 4-4}{\frac{6}{2}+5} \cdot 4^2-18\end{align*}
7. \begin{align*}6^2-[9+(7-5)^3]+49 \div 7\end{align*}
8. \begin{align*}\frac{27 \div 3^2+(6-2^2)}{(32 \div 8+1) \cdot 3}\end{align*}
9. \begin{align*}6+5 \cdot 2-9 \div 3+4\end{align*}
10. Using #18 above, insert parenthesis to make the expression equal 1. You may need to use more than one set of parenthesis.
11. Using #18 above, insert parenthesis to make the expression equal 23. You may need to use more than one set of parenthesis.

To see the Review answers, open this PDF file and look for section 1.3.

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Color Highlighted Text Notes

### Vocabulary Language: English

The additive inverse or opposite of a number x is -1(x). A number and its additive inverse always sum to zero.

Brackets

Brackets [ ], are symbols that are used to group numbers in mathematics. Brackets are the 'second level' of grouping symbols, used to enclose items already in parentheses.

Grouping Symbols

Grouping symbols are parentheses or brackets used to group numbers and operations.

Multiplicative Identity

The multiplicative identity for multiplication of real numbers is one.

Multiplicative Inverse

The multiplicative inverse of a number is the reciprocal of the number. The product of a real number and its multiplicative inverse will always be equal to 1 (which is the multiplicative identity for real numbers).

nested parentheses

Nested parentheses describe groups of terms inside of other groups. By convention, nested parentheses may be identified with other grouping symbols, such as the braces "{}" and brackets "[]" in the expression $\{ 3 + [ 2 - ( 5 + 4 ) ] \}$. Always evaluate parentheses from the innermost set outward.

Parentheses

Parentheses "(" and ")" are used in algebraic expressions as grouping symbols.

PEMDAS

PEMDAS (Please Excuse My Daring Aunt Sally) is a mnemonic device used to help remember the order of operations: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.

Real Number

A real number is a number that can be plotted on a number line. Real numbers include all rational and irrational numbers.

Simplify

To simplify means to rewrite an expression to make it as "simple" as possible. You can simplify by removing parentheses, combining like terms, or reducing fractions.