A 1.5 – Algebra Extension: the Order of Operations and Exponents

Kathleen Offenholley; Fatima Prioleau

A 1.5 – Algebra Extension: the Order of Operations and Exponents

Chapter 1, Section A5

Algebra Topics – The Order of Operations and Combining Rules for Exponents

Elementary Education – Build an expression with the largest value

The Order of Operations

Once you know all these different rules for exponents, you can combine them to simplify pretty complicated expressions. The key is to take them one step at a time, and to remember that each step has a reason — we’re not just randomly moving things around.

The first thing we need to know is what order to use to simplify things, and for that, we’ll use the Order of Operations. You may remember this as PEMDAS, Please Excuse My Dear Aunt Sally, BODMAS (if you grew up in Great Britain or a former British Colony), or some other way to remember them. The trouble with memory devices like these is that actually, there is a reason for the order we use — and the memory device hides that reason. It also hides a few other things about the order of operations that we will see below.

The order of operations is

Anything inside parenthesis or brackets ( ) is done first.
Next, anything raised to an exponent is calculated or simplified.
Multiplication and division are done next — but they are tied, multiplication is not first, instead, go from left to right.
Addition and subtraction last, again, they are tied, go from left to right.

The reason for this order is that it takes us from the most powerful to the least — from the operation that makes the biggest change to the one that makes the least. To illustrate this, look at what happens to the numbers 2 and 3 if we apply each operation — what happens to our answers?

2³ = 8

2 × 3 = 6

2 + 3 = 5

We go from the biggest answer to the smallest, using the order of operations.

It’s still okay to use a memory device to remember — for example, PEMDAS is Parenthesis, Exponents, Multiplication and Division, Addition and Subtraction — but don’t let that stop you from knowing that the order does (and should) make sense!

Example 1 Simplify $5 + 8 \div 4 \times 2^3 - 10$ using the correct order of operations.

There are no parenthesis, so the exponent is evaluated first. We have $5 + 8 \div 4 \times 2^3 - 10 = 5 + 8 \div 4 \times 8 - 10$ . Next, we do multiplication and division, going from left to right, so the division 8 ÷ 4 is next. This gives us $5 + 8 \div 4 \times 8 - 10 = 5 + 2 \times 8 - 10$ . Now we multiply, and then finally, do addition and subtraction from left to right. $5 + 2 \times 8 - 10 = 5 + 16 - 10 = 21 - 10 = 11$ . This is the same answer you will get if you enter the whole expression that we started with into a calculator.

Notice that none of the memory devices tell you that multiplication and division are tied, and that you go from left to right — they all make it seem as if multiplication comes first.

Question: Now you try!

Use your understanding of the order of operations to see if you can guess where to place the numbers below. The box on top right is an exponent. Compete against other people in class to see who can get the largest number!

In this activity, change the location of the parenthesis to get the largest value. Click on “Evaluate” to see the answer. Challenge! See if you can see why there is one best place to put the parentheses to make the largest value.

Rules for Exponents and the Order of Operations

In addition, here’s an interesting thing about the rules of exponents that might help you remember them — each rule goes down one level in the order of operations. Here’s how: Parenthesis, exponents raised to exponents with an arrow to multiplication multiplication and division with an arrow from multiplication to addition and an arrow from division to subtraction

When we raise exponential expressions to exponents, we multiply them.
When we multiply exponential expressions that have the same base, we add them.
When we divide exponential expressions that have the same base, we subtract them.

So now, knowing the order we should go in, let’s try a few examples.

Combining Rules for Exponents Using the Order of Operations

Example 2 Simplify $\frac{(2x^-^3y^5)^4}{10x^2y^8}$ .

This problem might look very complicated! But if we follow our order of operations, we can go through it step by step.

First, is there anything inside the parentheses that can be combined or simplified? We have $(2x^-^3y^5)$ inside parenthesis. None of these can be combined, since we have a coefficient and two different variables.
So now let’s go to exponents raised to exponents. We have an exponent, 4, outside the parentheses $(2x^-^3y^5)^4$ which means that everything inside the parentheses is being raised to the fourth power.

We can think of $(2x^-^3y^5)$ as our repeating factor: $(2x^-^3y^5)(2x^-^3y^5)(2x^-^3y^5)(2x^-^3y^5)$ , and then since it’s all multiplication, and multiplication can be done in any order, we can put the 2’s together to get 2 • 2 • 2 • 2 = 16, put the x’s together $(x^-^3)(x^-^3)(x^-^3)(x^-^3) = x^-^1^2$ and put the y’s together: $(y^5)(y^5)(y^5)(y^5) = y^2^0$ .

Or, since it’s all multiplication, we can think of each item inside the parentheses as being raised to the 4th power: $(2x^-^3y^5)^4 = 2^4 (x^-^3)^4 (y^5)^4 = 16 x^-^1^2 y^2^0$

3. Now we have $\frac{16 x^-^1^2 y^2^0}{10x^2y^8}$ and we can work on the division.

We could divide 16 ÷ 10 = 1.6, or we could simplify $\frac{16}{10}$ by dividing both by the common factor of 2, to get $\frac{8}{5}$ . When you do the homework for this section, be sure to check whether you are asked to write all answers in fraction form, or whether decimal form is allowed.

For the variables, we must subtract the exponents. For x, we get -12 – 2 = -14. For y, we get 20 – 8 = 12. This gives us: $\frac{8 x^-^1^4 y^1^2}{5}$ . We can stop there, or if we are asked to write the expression using only positive exponents, write $\frac{8 y^1^2}{5x^1^4}$ .

It is also possible to figure out the exponents by starting with $\frac{8 x^-^1^2 y^2^0}{5x^2y^8}$ and immediately taking the reciprocal of x^-12 by bringing it down to the denominator: $\frac{8 x^-^1^2 y^2^0}{5x^2y^8} = \frac{8 y^2^0}{5x^1^2x^2y^8}$ . We can then add the exponents 12+2 in the numerator to get $\frac{8 y^1^2}{5x^1^4}$ .

Question: Now you try!

Example 3 Simplify $\frac{(4a^-^3b^5)(3a^8b^4)}{10a^3b^1^0}$ .

Again, this seems as if it could be very difficult, but we can do it if we follow the order of operations step by step.

First, is there anything inside the parentheses that can be combined or simplified? No.
Are there any exponents raised to exponents. Again, no.
Let’s turn to multiplication and division. In the numerator, we have $(4a^-^3b^5)(3a^8b^4)$ , two expressions being multiplied. We can combine them by adding the exponents, but we still multiply the coefficients, 4 × 3. We get $(12a^-^3^+^8b^5^+^4) = 12a^5b^9$ . This gives us the fraction $\frac{12a^5b^9}{10a^3b^1^0}$ . Now we simplify or divide the coefficients, and subtract the exponents. When simplifying $\frac{12}{10}$ , divide both 12 and 10 by 2 to get $\frac{6}{5}$ . With the exponents, we have 5 – 3 =2, and 9 – 10 = -1. This gives us the fraction $\frac{6a^2b^-^1}{5}$
If the instructions say to write using only positive exponents, we rewrite our answer as $\frac{6a^2}{5b}$ . If decimals are allowed, we could also write $1.2 \frac{a^2}{b}$ .

Question: Now you try!

Summary of Rules of Exponents

Here are the rules we discovered in this section and the previous one:

Any expression (except for zero) to the zero power is 1. Written algebraically, $a^0=1$ , where a ≠ 0.

Tip: this includes the whole expression inside ( ). For example, $(3x^4y^5)^0 = 1$

Positive exponents indicate repeated multiplication. Written algebraically, $a^n=a\cdot a\cdot a\cdot a\ldots$ , where there are n factors of a.

Tip: if we have an expression in parenthesis raised to an exponent, the whole expression is repeated, so any coefficient inside the parenthesis ends up raised to that power. For example, $(2x)^3 = (2x) \cdot (2x) \cdot 2x = 8x^3$ .

Negative exponents indicate repeated division, Written algebraically, we have $a^-^n = \frac {1}{a^n}$ . If we have an expression in parenthesis raised to a negative exponent, the whole expression is divided.

Tip: If only part of the expression is raised to a negative exponent, only that part is divided. For example, $(5y)^-^3 = \frac {1}{5y^-^3}$ , but $5y^-^3 = \frac {5}{y^-^3}$ ; similarly, 4^-1is not -4, it is $\frac {1}{4}$

When multiplying exponential expressions that have the same base, add the exponents. Written algebraically, we have $a^m\cdot a^n = a^n^+^m$ .

Tip: if there are coefficients in front of the variables, they still get multiplied. For example, $3x^7\cdot 4x^2 = 12x^9$ .

When dividing exponential expressions that have the same base, subtract the exponents. Written algebraically, we have $\frac{a^m}{a^n} = a^n^-^m$ .

Tip: if there are coefficients in front of the variables, they still get divided. For example, $\frac{15x^7}{3x^2} = 5x^5$ .

When we have an exponential expression raised to another exponent, we multiply the exponents. Written algebraically, we have $(a^m)^n = a^n^m$ .

Tip: if there are coefficients in front of the variables, they still get raised to that exponent. For example, $(3x^4)^2 = 9x^8$

Again, as you’re getting used to these rules, it’s always okay to expand the expression to see what answer makes the most sense! It is so important that math should make sense, and that you do not just blindly follow the rules someone told you.

Video help can be found, here.

The homework below is a pdf of the online homework available in https://imathas.helpyourmath.com/. It does not include answers. Note that it is in two parts. In the first part, the homework contains images for each homework; the second part is text, to make the problems accessible to screen readers. Algebra extension homework A2

Unless otherwise specified, images in this section are by Kathleen Offenholley and licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

Chapter 1, Section A5

The Order of Operations

Example 1 Simplify using the correct order of operations.

Rules for Exponents and the Order of Operations

Combining Rules for Exponents Using the Order of Operations

Example 2 Simplify .

Example 3 Simplify .

Summary of Rules of Exponents

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Example 1 Simplify $5 + 8 \div 4 \times 2^3 - 10$ using the correct order of operations.

Example 2 Simplify $\frac{(2x^-^3y^5)^4}{10x^2y^8}$ .

Example 3 Simplify $\frac{(4a^-^3b^5)(3a^8b^4)}{10a^3b^1^0}$ .