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

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.”

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: ⓐ $4+3·7$ ⓑ $(4+3)·7.$

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.

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Simplify: $18÷6+4\left(5-2\right).$

Solution

Parentheses? Yes, subtract first.	$18÷6+4\left(5-2\right)$ $$
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.

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When there are multiple grouping symbols, we simplify the innermost parentheses first and work outward.

Simplify: $5+2^3+3\left[6-3\left(4-2\right)\right].$

Solution

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.

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Evaluate an expression

In the last few examples, we simplified expressions using the order of operations. Now we’ll evaluate some expressions—again following the order of operations. To evaluate an expression means to find the value of the expression when the variable is replaced by a given number.

To evaluate an expression, substitute that number for the variable in the expression and then simplify the expression.

Evaluate $7x-4,$ when ⓐ $x=5$ and ⓑ $x=1.$

Solution

ⓐ

Multiply.
Subtract.

ⓑ

Multiply.
Subtract.

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Evaluate $x=4,$ when ⓐ $x^2$ ⓑ $3^x.$

Solution

ⓐ

	$x^2$
Use definition of exponent.	$4·4$
Simplify.	$16$

ⓑ

	$3^{\text{x}}$
Use definition of exponent.	$3·3·3·3$
Simplify.	$81$

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Evaluate $2x^2+3x+8$ when $x=4.$

Solution

	$2x^2+3x+8$
Follow the order of operations.	$2\left(16\right)+3\left(4\right)+8$
	$32+12+8$
	$52$

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