Order of Operations

 Apply the rules in the right-hand frame in these examples.

Example 1: Simplify 10 - 42
Do we do the subtraction or the exponent first?

 The exponent, or power, is first.

Even though the subtraction occurs first, reading from left to right?

 Yes.

Apply the exponent.

 We get 10 - 16

Now do the subtraction.

 We get 10 - 16 = -6

Example 2: Simplify 52 + 22 - (3 + 1)2
Do we do the exponents first here?

 No. We need to do the operations inside the parentheses first.

Do the addition inside the parentheses.

 We get 52 + 22 - 42

Apply the exponents.

 We get 25 + 4 - 16

Why don't we get a "+ 16"?

 The exponent applies to only the group, variable, or number which precedes it. In this case, it applies to just the 4.

Complete the example.

 25 + 4 - 16 = 13.

Example 3: Simplify 2 + 5 - 3[4 - 2(8 - 5)]
Where do we start?

 Inside the grouping symbols.

Which set is first?

 We start with the innermost set.

Which set is that?

 We start with the parentheses.

Do that.

 We get 2 + 5 - 3[4 - 2(3)]

What operation is assumed between the "2" and the "(3)"?

 Multiplication.

Do it.

 We get 2 + 5 - 3[4 - 6]

What is next?

 Simplify inside the [ ].

Do it.

 We get 2 + 5 - 3[-2].

Do the multiplication and complete the example.

 We get 2 + 5 + 6 = 13

Example 4: Simplify (-2)(-4)(-3)
What is the operation assumed to be between the parentheses?

 Multiplication.

What will the algebraic sign of the result be?

Since we have the product of an odd number of negative quantities, the product will be negative.

Complete the problem.

 (-2)(-4)(-3) = -24

Example 5: Simplify (-2) - 4(-3). Carefully compare this to example 4. What do we do first?

The multiplication.

What will the algebraic sign of the multiplication be?

Since we have a positive 4 and a negative 3, we'll have a negative quantity to subtract.

Do the multiplication.

(-2) - 4(-3) = (-2) - (-12)

Complete the problem.

(-2) - (-12) = -2 + 12 = 10

Example 6: Simplify 2(-4 - 3)
What do we do first?

 We do the subtraction inside the parentheses.

Do it.

2(-4 - 3) = 2( -7 ) = -14

Example 7: Simplify -52
Does the exponent apply to "5" or to "-5"?

The exponent applies to the symbol or group which precedes it. Since there are no grouping symbols here, the exponent applies to just the "5".

What is the result?

 -52 = -(5)(5) = -25.

How could we express the need to square "-5" in some other problem?

We would have to use parentheses: (-5)2 = (-5)(-5) = 25

Example 8: Simplify 6 - 3[ 2 ( 5 - 2 ) + 4 ]
Where do we start here?

Inside the innermost grouping symbols.

Which ones are those?

The parentheses.

Do it.

 6 - 3[ 2 ( 5 - 2 ) + 4 ] = 6 - 3[ 2 ( 3 ) + 4 ]

Do the inner multiplication.

 6 - 3[ 2 ( 3 ) + 4 ] = 6 - 3[ 6 + 4 ]

Now what?

 Simplify inside the [ ].

Complete the problem.

6 - 3[ 6 + 4 ] = 6 - 3[ 10 ] = 6 - 30 = -24

Example 9: Simplify 6 ¸ 2 * 4
What do we do first here?

Item 2 in the right-hand frame applies. We do division and multiplication together, in order, from left to right.

So, do we divide first?

Yes.

Do it.

 6 ¸ 2 * 4 = 3 * 4 = 12

If we wanted the multiplication to be done first, how could we express that?

We could use parentheses and write 6 ¸ (2 * 4) = 6 ¸ 8 = 3/4.
What a different result!

Example 10: Simplify | 2 - 3 - 4 | + (-3 + 5)2
How do we handle the absolute value signs?

 We treat them as grouping symbols.

Simplify inside each group.

 | 2 - 3 - 4 | + (-3 + 5)2 = | 2 - 7 | + ( 2 )2
= | -5 | + 4

Evaluate the absolute value and complete the problem.

| -5 | + 4 = 5 + 4 = 9