1.2 Use the language of algebra (Page 4/18)

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Students often ask, “How will I remember the order?” Here is a way to help you remember: Take the first letter of each key word and substitute the silly phrase: “Please Excuse My Dear Aunt Sally.”

\begin{matrix} P arentheses & P lease \\ E xponents & E xcuse \\ M ultiplication D ivision & M y D ear \\ A ddition S ubtraction & A unt S ally \end{matrix}

It’s good that “ M y D ear” goes together, as this reminds us that m ultiplication and d ivision have equal priority. We do not always do multiplication before division or always do division before multiplication. We do them in order from left to right.

Similarly, “ A unt S ally” goes together and so reminds us that a ddition and s ubtraction also have equal priority and we do them in order from left to right.

Let’s try an example.

Simplify: ⓐ $4 + 3 \cdot 7$ ⓑ $(4 + 3) \cdot 7 .$

Solution

ⓐ


Are there any p arentheses? No.
Are there any e xponents? No.
Is there any m ultiplication or d ivision? Yes.
Multiply first.
Add.

ⓑ


Are there any p arentheses? Yes.
Simplify inside the parentheses.
Are there any e xponents? No.
Is there any m ultiplication or d ivision? Yes.
Multiply.

Parentheses? Yes, subtract first.	$18 \div 6 + 4 (5 - 2)$
Exponents? No.
Multiplication or division? Yes.
Divide first because we multiply and divide left to right.
Any other multiplication or division? Yes.
Multiply.
Any other multiplication or division? No.
Any addition or subtraction? Yes.


Are there any parentheses (or other grouping symbol)? Yes.
Focus on the parentheses that are inside the brackets.
Subtract.
Continue inside the brackets and multiply.
Continue inside the brackets and subtract.
The expression inside the brackets requires no further simplification.
Are there any exponents? Yes.
Simplify exponents.
Is there any multiplication or division? Yes.
Multiply.
Is there any addition or subtraction? Yes.
Add.
Add.

	$3^{x}$

Use definition of exponent.	$3 \cdot 3 \cdot 3 \cdot 3$
Simplify.	$81$

	$2 x^{2} + 3 x + 8$

Follow the order of operations.	$2 (16) + 3 (4) + 8$
	$32 + 12 + 8$
	$52$

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1.2 Use the language of algebra (Page 4/18)

Solution

Solution

Solution

Evaluate an expression

Evaluate an expression

Solution

Solution

Solution

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