Simplifying expressions and the skillset needed to accurately simplify expressions are essential building blocks in algebra. Throughout your algebra and more advanced math classes, you’ll be using these rules and techniques, so mastering the key steps in simplifying expressions will give you an edge later on.
In this article, we’ll break down the fundamentals of PEMDAS and refresh the different algebraic properties that will come in handy when simplifying expressions. You’ll have a chance to try different problems to also test your understanding. By the end of our discussion, you’ll feel confident to work on more complex expressions!
What Are the Steps in Simplifying Expressions?
When simplifying expressions, group appropriate terms together and apply the rules of operation in the correct order. Writing the final and simplified expression in its standard form is a great last step to follow. This means that there are different approaches when simplifying expressions, but here are some steps to help guide you:
Step 1: Eliminate the parentheses and brackets by evaluating, distributing, and combining terms inside them.
Step 2: Rewrite terms so that they share the same form, so evaluate terms with exponents and rewrite mixed numbers to fractions.
Step 3: Multiply and divide terms when needed and when these operations are present.
Step 4: Add and subtract terms from left to right to combine like terms.
Step 5: When you can no longer combine any terms, stop and rewrite the final expression in standard form.
Do these steps sound familiar? Because you’ve already encountered some of these rules in the form of the PEMDAS rule. Recall that PEMDAS is an arithmetic rule that stands for: Parenthesis, Exponents, Multiplication, Division, Addition, and Subtraction. It’s the general rule to follow when performing a series of arithmetic operations.
| Applying PEMDAS Rules in Arithmetic Operations |
| \begin{aligned}5\times (12 -8) + 3(5 -2) - 3^2 &= 5 \times {\color{DarkGreen}4} + 3\times{\color{DarkGreen}3} - 3^2\,\,,\color{DarkGreen}\text{Parenthesis}\\&= 5\times 4 + 3\times 3 - {\color{DarkGreen}9}\,\,,\color{DarkGreen}\text{Exponent}\\&={\color{DarkGreen}20} + {\color{DarkGreen}9} - 9\,\,,\color{DarkGreen}\text{Multiplication}\\&= {\color{DarkGreen}29} - 9\,\,,\color{DarkGreen}\text{Addition and Subtraction}\\&=20\end{aligned} |
Take a look at how the expression shown above are evaluated and simplified to appreciate the PEMDAS rule even better. Aside from following the PEMDAS closely when simplifying expressions, it’s important to learn how to identify like terms. Recall that like terms share the same variable and power. Here are some examples to help you refresh your skill in identifying like terms and unlike terms:
| Like Terms | Unlike Terms |
| 4x 6x | 2x 5x^2 |
| -5mn 10mn | -5mn^2 -5m^2n |
| 2a^2b 3a^2b | 4ab^2 3a^3b |
| \dfrac{1}{2}x^3y^2 \dfrac{2}{3}x^3y^2 | \dfrac{1}{5}xy^2z \dfrac{1}{5}x^2y^3z^2 |
Like terms must share both variable and power. It’s essential that both of these conditions are met when combining like terms. This is crucial when simplifying algebraic expressions, so let us show you an example of how these rules are applied with algebraic terms.
| Applying PEMDAS Rules With Algebraic Terms |
| \begin{aligned}3mn - 2(4mn + n) + 6n&= 3mn {\color{DarkOrange} -8mn - 2n} + 6n\,\,,\color{DarkOrange}\text{Parenthesis}\\&= {\color{DarkOrange}(3mn - 8mn)} + {\color{DarkOrange}(-2n + 6n)}\,\,,\color{DarkOrange}\text{Group Terms}\\&={\color{DarkOrange}-5mn} + {\color{DarkOrange}4n}\,\,,\color{DarkOrange}\text{Add and Subtract}\\&= 4n - 5mn\end{aligned} |
There are also different rules to remember when simplifying expressions and you’d learn more about them in the next section. For now, test your understanding by answering the problems shown below.
Problem 1
To simplify the expression, 3(4x – 5y) + 2x – 6y + 4^2, what should you do first?
A. Rewrite 42 into its whole number counterpart, 16.
B. Group all like terms by inspecting the variable and power shared.
C. Group the last three terms using a bracket.
D. Distribute 3 to eliminate the parenthesis first.
Recall that when simplifying algebraic expressions, the PEMDAS rule still applies. In PEMDAS, the first step is to eliminate existing brackets and parenthesis by distributing and simplifying terms. For 3(4x – 5y) + 2x – 6y + 4^2, the grouped terms are enclosed by a parenthesis with a constant factor of 3.
This means that when simplifying the given algebraic expression, begin by eliminating the parenthesis first. Hence, the first step is to distribute 3 to each of the term inside the parenthesis making D the correct answer.
Problem 2
Identify whether each pair of terms are like terms or unlike terms.
a. 12mn, 10mn
When determining whether two terms are like terms or not, inspect the variables and the power of these variables for each of these terms.
| Term | Variable | Power |
| 12mn | m | n |
| 1 | 1 | |
| 10mn | m | n |
| 1 | 1 |
From these, we can see that both terms share the same variables and each variable share the same power. Hence, the two terms are like terms.
b. 2x^2y, 3xy^2
Using a similar process, inspect the terms’ variables and powers to see whether the two are like terms or unlike terms.
| Term | Variable | Power |
| 2x^2y | x | y |
| 2 | 1 | |
| 3xy^2 | x | y |
| 1 | 2 |
Taking a look at the table, it’s easy to see that they both share the same variables. However, each variable do not share the same power. Remember that for two terms to be like terms, each of the variables must share the same power. Hence, the two terms are unlike terms.
After simplifying an expression and you end up with unlike terms, that’s the sign that you’ve reached near the end of the process. An additional step would be to rewrite the resulting expression in standard form, if any, to make it easier to inspect or review later on.
At this point, you’ve probably encountered the terms distribute, fractions, and even exponents throughout this discussion. That’s because the process of simplifying expressions utilizes a wide range of rules and properties that you’ve learned in the past. Depending on the complexity of the expression that needs to be simplified, you’ll need to ensure that you’re comfortable utilizing these rules.
What Are Additional Rules for Simplifying Expressions?
There are additional rules that will help you in simplifying expressions and some of them are:
- Proper way to add and subtract like terms using their coefficients.
- Applying the distributive property accurately.
- When multiplying a negative coefficient, ensure that the terms’ signs also change.
- Ensure the proper application of the rules of exponents.
Remember these pointers when simplifying expressions. This section will show you different examples of where these rules apply. For now, take a quick refresher on how to combine like terms by adding or subtracting their coefficients.
Simplifying Expressions With Like Terms
When simplifying expressions with like terms, add or subtract the like terms’ coefficients. For example, when working with 4mn + 6mn in one expression, add the two terms by adding their coefficients and retaining the variable part.
\begin{aligned}{\color{Purple}4}mn + {\color{Purple}6}mn &= {\color{Purple}(4+ 6)}mn\\&= 10mn\end{aligned}Similarly, when subtracting two terms such as 5a^2b – 2a^2b, retain the variable part and subtract the terms’ coefficients.
\begin{aligned}{\color{Purple}5}a^2b - {\color{Purple}2}a^2b &= {\color{Purple}(5 - 2)}a^2b\\&= 3a^2b\end{aligned}These are steps that you’ll often encounter when simplifying expressions, so it’s important that you’re comfortable with combining like terms.
Problem 3
Simplify the following expressions by combining like terms.
a. 5x^2y + 6x^2y
By inspecting the variables and their powers, we can see that the two terms are like terms. When combining like terms, add (or subtract) the terms by retaining the shared variables and adding (or subtracting) the coefficients.
\begin{aligned}{\color{Teal}5}x^2y + {\color{Teal}6}x^2y &= {\color{Teal}(5 + 6)}x^2y\\&=11x^2y\end{aligned}By adding the terms’ coefficients, we have added and simplified the two like terms into one simplified term. The expression, 5x^2y + 6x^2y, can still be simplified to 11x^2y.
b. 12ab + 3mn – 6ab + 2mn
When asked to simplify terms with three or more terms, always group like terms altogether first.
\begin{aligned}{\color{DarkBlue}12ab} + {\color{DarkRed}3mn} {\color{DarkBlue}-6ab} + {\color{DarkRed}2mn}&= {\color{DarkBlue}(12ab - 6ab)} + {\color{DarkRed}(3mn + 2mn)} \end{aligned}Now that the like terms have been grouped together, simplify each group of terms by applying the same process. Focus on the coefficients for each pair of like terms when simplifying them.
\begin{aligned}{\color{DarkBlue}(12ab - 6ab)} + {\color{DarkRed}(3mn + 2mn)} &= {\color{DarkBlue}(12 - 6)ab} + {\color{DarkRed}(3 +2)mn}\\&= 6ab + 5mn\end{aligned}Since the resulting terms no longer share the same variables and powers, there is no need to simplify the expression further. Hence, the simplified form of 12ab + 3mn – 6ab + 2mn is equal to 6ab + 5mn.
Simplifying Expressions With Distributive Property
Simplifying expressions with distributive property is usually the first step you do when the expression contains coefficients before brackets or parenthesis. When distributing the coefficient or factor into the parenthesis, multiply each terms inside the parenthesis by the coefficient.
\begin{aligned}k(a + b) &= k\cdot a + k\cdot b\\k(a - b) &= k\cdot a - k\cdot b\end{aligned}If the coefficient is negative, account for its negative sign and make sure to change the signs of the terms inside the parenthesis. After distributing any existing factors, remove the parenthesis enclosing each term.
\begin{aligned}4(3x + 2y) &={\color{Orchid}4}(3x) +{\color{Orchid}4}(2y)\\&=12x +8y\\\\-2(5a - b) &={\color{Orchid}-2}(5a) +{\color{Orchid}-2}(-b)\\&=-10a + 2b\end{aligned}Now that you’ve had a quick refresher on how to apply the distributive property, why don’t you try simplifying the expression shown below?
Problem 4
Simplify the following expression, 2(x – m) + 4(x + m), using the different rules that you’ve learned.
When working with expressions with parenthesis or two, eliminate them first. When you see coefficients before the parenthesis, such as 2 and 4, apply the distributive property.
\begin{aligned}2(x – m) + 4(x + m)&={\color{Orchid}2}(x) -{\color{Orchid}2}(m) +{\color{Teal}4}(x) +{\color{Teal}4}(m)\\&= 2x - 2m + 4x + 4m\end{aligned}By inspection, we can still that there are like terms that can still be combined. Group the like terms and evaluate them to simplify the expression further.
\begin{aligned}2x - 2m + 4x + 4m &= {\color{Orchid}(4m - 2m)} + {\color{Teal}(2x +4x)}\\&={\color{Orchid}(4-2)}m + {\color{Teal}(2 + 4)}x\\&= 2m + 6x\end{aligned}By applying the correct properties and rules, 2(x – m) + 4(x + m) has been simplified to 2m + 6x.
Simplifying Expressions With Exponents
When simplifying expressions with exponents, it’s important to follow the rules of exponents accurately. After eliminating the parenthesis, the next step in simplifying expressions is to evaluate terms with exponents. This is why it’s important that you’re familiar with the fundamental rules of exponents.
| Rules of Exponents | |
| Zero Rule | b^0 = 1 |
| Identity Rule | b^1 = b |
| Negative Exponent Rule | b^{-m} = 1/ b^m |
| Product Rule | b^m \cdot b^n = b^{m +n} |
| Quotient Rule | b^m/b^n = b^{m –n} |
| Power of a Power Rule | (b^m)^n = b^{mn} |
| Power of a Product Rule | (ab)^m = a^mb^m |
| Power of a Quotient Rule | (a/b)^m = a^m/ b^m |
Apply these rules when simplifying expressions with exponents. Of course, eliminating the parenthesis is still the first step to work on when you have to.
Problem 5
Simplify the following expressions by applying the appropriate rules.
a.
4 + 2(1/4 + 1/2) – (1/2)^2 +2^3Of course, begin by eliminating the parenthesis by distributing 2 into the terms contained by the parenthesis. Then evaluate the terms with exponents by applying the appropriate rules of exponents.
\begin{aligned}4 + 2(1/4 + 1/2) – (1/2)^2 +2^3 &= 4 + {\color{DarkOrange}2}(1/4) +{\color{DarkOrange}2}(1/2)– (1/2)^2 +2^3\\&=4 + 1/2 + 1-(1/2)^2 +2^3\\&= 4 + 1/2 + 1{\color{DarkOrange}-1/4 + 8}\end{aligned}Now, simplify the expression further by adding the terms from left to right.
\begin{aligned}4 + 1/2 + 1-(1/2)^2 +2^3&= 4 + 1/2 + 1-1/4 + 8\\&=9/2+ 1-1/4 + 8\\&=11/2 -1/4 +8\\&=21/4 +8\\&=53/4\end{aligned}This means that we can simplify the expression to 53/4.
b. (ab)^2 + 2(a^2b^2 + m^2n^2) - m^2n^2
Now, apply a similar process to simplify the algebraic expression. Distribute 2 into the parenthesis then evaluate the first term by applying the power of a product rule.
\begin{aligned}(ab)^2 + 2(a^2b^2 + m^2n^2) - m^2n^2&= (ab)^2 + {\color{Purple}2}(a^2b^2)+{\color{Purple}2}(m^2n^2) - m^2n^2\\&=(ab)^2 + 2a^2b^2 +2m^2n^2-m^2n^2\\&={\color{Purple}a^2b^2}+ 2a^2b^2 +2m^2n^2-m^2n^2\end{aligned}Inspect for like terms (check the variable and make sure they also share the same powers). Group these terms then combine the like terms.
\begin{aligned}a^2b^2+ 2a^2b^2 +2m^2n^2-m^2n^2&={\color{Purple}(a^2b^2+ 2a^2b^2)} +{\color{Teal} (2m^2n^2-m^2n^2)}\\&= {\color{Purple}(1 + 2)}a^2b^2 + {\color{Teal}(2 -1)}m^2n^2\\&=3a^2b^2+m^2n^2\end{aligned}Now that there are no more like terms that can be found from the simplified expression, we can now stop. Hence, the simplified expression is now 3a^2b^2+m^2n^2.
These examples have shown that by applying the correct properties and algebraic rule in the right order, simplifying expressions won’t be as intimidating. By making a systematic approach, you avoid making crucial arithmetic mistakes. Review this article whenever you need to and practice the problems again if you want to!