"For -3 to the fourth power, how does it end up staying as a negative? I thought two negatives, when multiplied, became positive.

You have to understand the difference of two things, if you have -3^4, order of operations require you to do the power before adding the negative, so you end up with a negative. You are thinking about (-3)*4 where you would be correct, 4 negatives would make a positive. Be careful about having or not having parentheses which give two opposite sign answers.

If you're wondering about the order of operations, than you have to remember G.E.M/D.A/S to remember the orders. First, (G)roupings (which could be parenthesis, brackets or anything of that sort). Then, it's (E)xponents (the mini numbers on top righthand corner of the numbers, like 5³, which makes you do 5 ∘ 5 ∘ 5). Next, it's (M)ultiplication or (D)ivision (I recommend doing whichever is first). Lastly, it's (A)ddition or (S)ubtraction (I recommend doing whichever is first). Hope this helped.

Main content

7th grade

Course: 7th grade > Unit 5

Lesson 8: Order of operations

Order of operations with rational numbers

Google Classroom

Let's go deeper with the order of operations. How do negation, absolute value, reciprocals fit into the order of operations? How do other properties like the distributive property or commutative property of addition relate?

Order of operations review

When there are lots of operations in an expression, we need to agree on which to evaluate first. That way, we will all agree that the expression has the same value. We could just use parentheses to always show what to do first, but we like to write things shorter when we can.

So instead, we agree to evaluate more powerful operations sooner. Multiplication is repeated addition, so we multiply before we add. Powers are repeated multiplication, so we take powers before we multiply.

Grouping symbols: We evaluate what is inside of grouping symbols first. There are lots of grouping symbols. Some common ones are parentheses, fraction bars, and absolute value symbols.
Exponents: Next we evaluate powers. There are a couple of operations that undo exponents (it takes $2$ ‍ operations because powers are not commutative). They happen in this step, too.
Multiplication: Then we multiply. Division undoes multiplication, so it happens in the same step.
Addition: Last of all, we add. Subtraction undoes addition, so it happens in the same step.

Fun fact: There are operations that grow a number faster than exponents. One of them shows up a lot in probability.

Can you think of any operations that change a number more slowly than addition? Tell us about it in the comments.

Exponents with negatives

How do negative signs fit in with order of operations? In the expression

- 4^{2}

, do we apply the negative sign or the exponent first? A negative sign means we take the opposite of a number. That's the same as multiplying the number by

- 1

. So we take the opposite in the multiplication and division step.

Let's evaluate

(- 4)^{2}

and

- 4^{2}

\begin{aligned} (- 4)^{2} & = - 4 \cdot (- 4) & Evaluate groups. \\ = 16 & Multiply. \end{aligned}

With

(- 4)^{2}

, we took the opposite of

4

first, because the negative sign was inside the grouping symbols.

\begin{aligned} - 4^{2} & = - (4 \cdot 4) & Evaluate the power. \\ = - 16 & Take the opposite. \end{aligned}

With

- 4^{2}

, we squared

4

first, because exponents come earlier in the order operations than multiplying by

- 1

does.

Problem 1.1

Evaluate.

- 3^{4} =

Exponents with fractions

Recall that fractions have three grouping symbols built in: the numerator, the denominator, and the entire fraction. If there are no parentheses or other grouping symbols visible, then the exponent is inside the numerator or denominator.

For example, in the expression

{(\frac{3}{2})}^{4}

, the entire fraction is inside a grouping symbol, the parentheses. The exponent

4

is outside the parentheses. So we are raising the entire fraction to the power of

3

\begin{aligned} {(\frac{3}{2})}^{4} & = \frac{3}{2} \cdot \frac{3}{2} \cdot \frac{3}{2} \cdot \frac{3}{2} \\ = \frac{81}{16} \end{aligned}

On the other hand, in the expression

\frac{3^{4}}{2}

, the only grouping symbol is the fraction bar. The exponent

4

is in the numerator. So we only raise the

3

to the power of

4

\begin{aligned} \frac{3^{4}}{2} & = \frac{3 \cdot 3 \cdot 3 \cdot 3}{2} \\ = \frac{81}{2} \end{aligned}

Problem 2.1

Evaluate.

\frac{2}{7^{2}} =

Absolute value

Suppose we have the expression

4 - 12 \cdot | - 7 \cdot - 11 - 100 |

. We know we evaluate the multiplication before the subtraction, but when do we take the absolute value?

First of all, the absolute value symbols are a grouping symbol. So stuff inside the symbols happens at the same step as other grouping symbols. So first, we multiply

- 7 \cdot - 11

to get

77

, then subtract

100

to get

- 23

inside of the absolute value symbols.

Then we take the absolute value during the same step as we apply exponents.

[Why?]

Fun fact: There are other operations that go at this same step because they undo powers the way subtraction undoes addition.

Problem 3.1

Evaluate.

4 - 12 \cdot | - 7 \cdot - 11 - 100 | =

Add and subtract left to right?

Maybe you learned that you should evaluate addition and subtraction from left to right. That's because subtraction does not commute or associate. Luckily, we can rewrite subtraction as addition of the opposite.

Let's rewrite

- 4 + 6 - 3 + 1

- 4 + 6 - 3 + 1 = \underset{term}{\underset{⏟}{- 4}} + \underset{term}{\underset{⏟}{6}} + (\underset{term}{\underset{⏟}{- 3}}) + \underset{term}{\underset{⏟}{1}}

Now that we're adding, we can change the order and groups!

Once we get used to thinking of subtracting as adding the opposite, we can take shortcuts. Often, people leave out the extra

+

symbols.

\underset{term}{\underset{⏟}{- 4}} + \underset{term}{\underset{⏟}{6}} \underset{term}{\underset{⏟}{- 3}} + \underset{term}{\underset{⏟}{1}}

Each time we reach a

+

-

that's not in a grouping symbol, it's the start of a new term. The

-

stays with its term. Then we can commute and associate any terms we want.

\begin{aligned} - 4 + 6 - 3 + 1 & = 6 + 1 - 3 - 4 \\ = (6 + 1) + (- 3 - 4) \\ = 7 + (- 7) \end{aligned}

Problem 4.1

Which $2$ ‍ of the following expressions are equivalent to $- 0.25 + (- 8) - 13 + 0.7$ ‍?

Multiply and divide left to right?

The reciprocal of a number tells us how many groups of it we can make from

1

. The great thing about reciprocals is that we can rewrite dividing by a number as multiplying by its reciprocal.

[Why?]

Here are a few examples.

$8 \div 6 = 8 \cdot \frac{1}{6}$ ‍
$\frac{3}{7} \div \frac{6}{11} = \frac{3}{7} \cdot \frac{11}{6}$ ‍
$\begin{aligned} 11 \div 0.25 & = 11 \div \frac{1}{4} \\ = 11 \times 4 \end{aligned}$ ‍

That's great news, because we can commute and associate multiplication factors in any order.

Let's rewrite

3 \div 8 \cdot - 24

\begin{aligned} 3 \div 8 \cdot - 24 & = 3 \cdot \frac{1}{8} \cdot - 24 \\ = 3 \cdot - 24 \cdot \frac{1}{8} \\ = 3 \cdot (- 24 \cdot \frac{1}{8}) \\ = 3 \cdot (- 24 \div 8) \end{aligned}

Like with adding and subtracting, sometimes we take shortcuts and move the division symbol with the implied factor.

\underset{factor}{\underset{⏟}{3}} \underset{factor of \frac{1}{8}}{\underset{⏟}{\div 8}} \cdot \underset{factor}{\underset{⏟}{- 24}} = 3 \cdot - 24 \div 8

We cannot start an expression with a

\div

symbol. So if we move the division to the start of an expression or grouping symbol, then we must write it as a factor of the reciprocal.

\underset{factor}{\underset{⏟}{3}} \underset{factor of \frac{1}{8}}{\underset{⏟}{\div 8}} \cdot \underset{factor}{\underset{⏟}{- 24}} = \frac{1}{8} \cdot 3 \cdot - 24

Note, we cannot commute a factor into a different term. Whenever we see a

+

-

symbol that is not inside a grouping symbol, that's the start of a new term. For example, the expression

10 \cdot 2 - 5

has

2

terms:

10 \cdot 2

and

- 5

\begin{aligned} 10 \cdot 2 - 5 & \neq 10 - 5 \cdot 2 \\ 20 - 5 & \neq 10 - 10 \\ 15 & \neq 0 \end{aligned}

Problem 5.1

Which $2$ ‍ of the following expressions are equivalent to $\frac{3}{4} \div \frac{8}{5} \div 6$ ‍?

Grouping symbols or distributing first

Recall that the distributive property allows us to write equivalent expressions involving parentheses and multiplication. So we get the same value whether we distribute first or evaluate what's inside the parentheses first.

Let's try it out.

\begin{aligned} 12 + (- 7) - (- 20 + 5) \\ = & 12 + (- 7) - (- 15) & Evaluate the group. \\ = & 12 + (- 7) + 15 & Multiply - 1 and - 15 . \\ = & 20 & Add. \end{aligned}

Now we'll distribute first.

\begin{aligned} 12 + (- 7) - (- 20 + 5) \\ = & 12 + (- 7) + 20 - 5 & Distribute - 1 . \\ = & 20 & Add. \end{aligned}

We get the same value both ways.

Note, the distributive property does not apply to all grouping symbols, just to parentheses.

\begin{aligned} - 7 | 3 - 4 | & \neq | - 7 (3) + (- 7) (- 4) | \\ - 7 & \neq 7 \end{aligned}

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