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Lesson 4: Order of operations

# Order of operations with rational numbers

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 $\left(-4{\right)}^{2}$ and $-{4}^{2}$.
$\begin{array}{rlr}\left(-4{\right)}^{2}& =-4\cdot \left(-4\right)& \text{Evaluate groups.}\\ \\ & =16& \text{Multiply.}\end{array}$
With $\left(-4{\right)}^{2}$, we took the opposite of $4$ first, because the negative sign was inside the grouping symbols.
$\begin{array}{rlr}-{4}^{2}& =-\left(4\cdot 4\right)& \text{Evaluate the power.}\\ \\ & =-16& \text{Take the opposite.}\end{array}$
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 ${\left(\frac{3}{2}\right)}^{4}$, 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 $4$.
$\begin{array}{rl}{\left(\frac{3}{2}\right)}^{4}& =\frac{3}{2}\cdot \frac{3}{2}\cdot \frac{3}{2}\cdot \frac{3}{2}\\ \\ & =\frac{81}{16}\end{array}$
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 $4$.
$\begin{array}{rl}\frac{{3}^{4}}{2}& =\frac{3\cdot 3\cdot 3\cdot 3}{2}\\ \\ & =\frac{81}{2}\end{array}$
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.
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 $-4+6-3+1$.
$-4+6-3+1=\underset{\text{term}}{\underset{⏟}{-4}}+\underset{\text{term}}{\underset{⏟}{6}}+\left(\underset{\text{term}}{\underset{⏟}{-3}}\right)+\underset{\text{term}}{\underset{⏟}{1}}$
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.
$\underset{\text{term}}{\underset{⏟}{-4}}+\underset{\text{term}}{\underset{⏟}{6}}\underset{\text{term}}{\underset{⏟}{-3}}+\underset{\text{term}}{\underset{⏟}{1}}$
Each time we reach a $+$ or $-$ 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.
$\begin{array}{rl}-4+6-3+1& =6+1-3-4\\ \\ & =\left(6+1\right)+\left(-3-4\right)\\ \\ & =7+\left(-7\right)\end{array}$
Problem 4.1
Which $2$ of the following expressions are equivalent to $-0.25+\left(-8\right)-13+0.7$?

## 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.
Here are a few examples.
• $8÷6=8\cdot \frac{1}{6}$
• $\frac{3}{7}÷\frac{6}{11}=\frac{3}{7}\cdot \frac{11}{6}$
• $\begin{array}{rl}11÷0.25& =11÷\frac{1}{4}\\ \\ & =11×4\end{array}$
That's great news, because we can commute and associate multiplication factors in any order.
Let's rewrite $3÷8\cdot -24$.
$\begin{array}{rl}3÷8\cdot -24& =3\cdot \frac{1}{8}\cdot -24\\ \\ & =3\cdot -24\cdot \frac{1}{8}\\ \\ & =3\cdot \left(-24\cdot \frac{1}{8}\right)\\ \\ & =3\cdot \left(-24÷8\right)\end{array}$
Like with adding and subtracting, sometimes we take shortcuts and move the division symbol with the implied factor.
We cannot start an expression with a $÷$ 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.
Note, we cannot commute a factor into a different term. Whenever we see a $+$ or $-$ 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 $-5$.
$\begin{array}{rl}10\cdot 2-5& \ne 10-5\cdot 2\\ \\ 20-5& \ne 10-10\\ \\ 15& \ne 0\end{array}$
Problem 5.1
Which $2$ of the following expressions are equivalent to $\frac{3}{4}÷\frac{8}{5}÷6$?

## 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.
Now we'll distribute first.
We get the same value both ways.
Note, the distributive property does not apply to all grouping symbols, just to parentheses.
$\begin{array}{rl}-7|3-4|& \ne |-7\left(3\right)+\left(-7\right)\left(-4\right)|\\ \\ -7& \ne 7\end{array}$

## Want to join the conversation?

• I found it helps when I read through the parts I don't understand a few times, and read them slowly. I try saying them out loud to find out how it makes sense.
• Fun fact: There are operations that grow a number faster than exponents

It's a factorial, right?
But I can't figure out this:
Can you think of any operations that change a number more slowly than addition?

I could make an addition very slow, like f(x)=x+0.0001, but something slower than addition? Could someone point a direction, please?
• If "slow" means very little difference, then yes, it just depends on how you utilize it. Multiplication can potentially be slower than addition.
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For example, if you multiply 5 with 1.0001, the product will be 5.0005, which is evidently less difference (with 0.0005) than if you added 5 and 1.0001, which would be 6.0001, which has a difference of 1.0001. When put on a graph, you can see that multiplication has the least difference when used this way. However, this only works when used in specific ways.
• i dont understand the fractions
• what does Pemdas mean
• Parentheses exponent multiply or divide and addition or subtraction
• Regarding an operation that change a number more slowly than addition; if exponents grow a number faster than multiplication and addition, might taking roots change a number slower than addition?
• Why is the question so hard for me?
• what is 9+6 to the power of two minas 2.8