Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can appear to be scary for beginner students in their early years of college or even in high school.
However, grasping how to process these equations is important because it is foundational knowledge that will help them navigate higher arithmetics and complicated problems across multiple industries.
This article will share everything you should review to know simplifying expressions. We’ll review the principles of simplifying expressions and then validate our comprehension through some practice problems.
How Does Simplifying Expressions Work?
Before you can learn how to simplify them, you must understand what expressions are at their core.
In arithmetics, expressions are descriptions that have no less than two terms. These terms can combine numbers, variables, or both and can be connected through addition or subtraction.
As an example, let’s go over the following expression.
8x + 2y - 3
This expression combines three terms; 8x, 2y, and 3. The first two terms contain both numbers (8 and 2) and variables (x and y).
Expressions that incorporate coefficients, variables, and occasionally constants, are also known as polynomials.
Simplifying expressions is important because it opens up the possibility of grasping how to solve them. Expressions can be written in convoluted ways, and without simplifying them, everyone will have a tough time attempting to solve them, with more chance for error.
Undoubtedly, all expressions will be different regarding how they are simplified based on what terms they incorporate, but there are general steps that apply to all rational expressions of real numbers, whether they are logarithms, square roots, etc.
These steps are called the PEMDAS rule, short for parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule shows us the order of operations for expressions.
Parentheses. Simplify equations between the parentheses first by using addition or subtracting. If there are terms right outside the parentheses, use the distributive property to apply multiplication the term on the outside with the one on the inside.
Exponents. Where workable, use the exponent properties to simplify the terms that have exponents.
Multiplication and Division. If the equation necessitates it, use multiplication and division to simplify like terms that apply.
Addition and subtraction. Then, use addition or subtraction the simplified terms of the equation.
Rewrite. Ensure that there are no additional like terms that need to be simplified, and then rewrite the simplified equation.
Here are the Rules For Simplifying Algebraic Expressions
Beyond the PEMDAS sequence, there are a few additional rules you must be informed of when dealing with algebraic expressions.
You can only apply simplification to terms with common variables. When adding these terms, add the coefficient numbers and leave the variables as [[is|they are]-70. For example, the expression 8x + 2x can be simplified to 10x by adding coefficients 8 and 2 and retaining the x as it is.
Parentheses that include another expression directly outside of them need to utilize the distributive property. The distributive property allows you to simplify terms on the outside of parentheses by distributing them to the terms on the inside, as shown here: a(b+c) = ab + ac.
An extension of the distributive property is called the property of multiplication. When two stand-alone expressions within parentheses are multiplied, the distributive property is applied, and each unique term will need to be multiplied by the other terms, resulting in each set of equations, common factors of each other. Like in this example: (a + b)(c + d) = a(c + d) + b(c + d).
A negative sign directly outside of an expression in parentheses indicates that the negative expression must also need to be distributed, changing the signs of the terms inside the parentheses. For example: -(8x + 2) will turn into -8x - 2.
Similarly, a plus sign right outside the parentheses means that it will be distributed to the terms inside. However, this means that you can remove the parentheses and write the expression as is owing to the fact that the plus sign doesn’t alter anything when distributed.
How to Simplify Expressions with Exponents
The prior properties were straight-forward enough to use as they only dealt with principles that affect simple terms with numbers and variables. Despite that, there are more rules that you must implement when dealing with expressions with exponents.
In this section, we will review the properties of exponents. Eight rules affect how we deal with exponents, those are the following:
Zero Exponent Rule. This rule states that any term with a 0 exponent is equivalent to 1. Or a0 = 1.
Identity Exponent Rule. Any term with a 1 exponent won't alter the value. Or a1 = a.
Product Rule. When two terms with the same variables are multiplied by each other, their product will add their two exponents. This is expressed in the formula am × an = am+n
Quotient Rule. When two terms with matching variables are divided, their quotient applies subtraction to their applicable exponents. This is expressed in the formula am/an = am-n.
Negative Exponents Rule. Any term with a negative exponent equals the inverse of that term over 1. This is written as the formula a-m = 1/am; (a/b)-m = (b/a)m.
Power of a Power Rule. If an exponent is applied to a term already with an exponent, the term will end up having a product of the two exponents applied to it, or (am)n = amn.
Power of a Product Rule. An exponent applied to two terms that possess differing variables should be applied to the respective variables, or (ab)m = am * bm.
Power of a Quotient Rule. In fractional exponents, both the denominator and numerator will assume the exponent given, (a/b)m = am/bm.
How to Simplify Expressions with the Distributive Property
The distributive property is the rule that states that any term multiplied by an expression within parentheses should be multiplied by all of the expressions on the inside. Let’s witness the distributive property used below.
Let’s simplify the equation 2(3x + 5).
The distributive property states that a(b + c) = ab + ac. Thus, the equation becomes:
2(3x + 5) = 2(3x) + 2(5)
The result is 6x + 10.
Simplifying Expressions with Fractions
Certain expressions contain fractions, and just as with exponents, expressions with fractions also have multiple rules that you need to follow.
When an expression includes fractions, here's what to remember.
Distributive property. The distributive property a(b+c) = ab + ac, when applied to fractions, will multiply fractions separately by their denominators and numerators.
Laws of exponents. This tells us that fractions will typically be the power of the quotient rule, which will apply subtraction to the exponents of the numerators and denominators.
Simplification. Only fractions at their lowest form should be expressed in the expression. Apply the PEMDAS rule and be sure that no two terms share matching variables.
These are the same properties that you can apply when simplifying any real numbers, whether they are binomials, decimals, square roots, logarithms, linear equations, or quadratic equations.
Sample Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this case, the principles that should be noted first are the PEMDAS and the distributive property. The distributive property will distribute 4 to all other expressions inside the parentheses, while PEMDAS will decide on the order of simplification.
Due to the distributive property, the term outside of the parentheses will be multiplied by the individual terms inside.
4(2x) + 4(5x) + 4(7) - 3y
8x + 20x + 28 - 3y
When simplifying equations, you should add all the terms with matching variables, and all term should be in its lowest form.
28x + 28 - 3y
Rearrange the equation as follows:
28x - 3y + 28
Example 2
Simplify the expression 1/3x + y/4(5x + 2)
The PEMDAS rule expresses that the the order should start with expressions inside parentheses, and in this case, that expression also needs the distributive property. In this scenario, the term y/4 should be distributed amongst the two terms within the parentheses, as follows.
1/3x + y/4(5x) + y/4(2)
Here, let’s put aside the first term for now and simplify the terms with factors associated with them. Remember we know from PEMDAS that fractions require multiplication of their numerators and denominators individually, we will then have:
y/4 * 5x/1
The expression 5x/1 is used for simplicity as any number divided by 1 is that same number or x/1 = x. Thus,
y(5x)/4
5xy/4
The expression y/4(2) then becomes:
y/4 * 2/1
2y/4
Thus, the overall expression is:
1/3x + 5xy/4 + 2y/4
Its final simplified version is:
1/3x + 5/4xy + 1/2y
Example 3
Simplify the expression: (4x2 + 3y)(6x + 1)
In exponential expressions, multiplication of algebraic expressions will be used to distribute every term to each other, which gives us the equation:
4x2(6x + 1) + 3y(6x + 1)
4x2(6x) + 4x2(1) + 3y(6x) + 3y(1)
For the first expression, the power of a power rule is applied, which means that we’ll have to add the exponents of two exponential expressions with the same variables multiplied together and multiply their coefficients. This gives us:
24x3 + 4x2 + 18xy + 3y
Because there are no other like terms to simplify, this becomes our final answer.
Simplifying Expressions FAQs
What should I remember when simplifying expressions?
When simplifying algebraic expressions, bear in mind that you must follow PEMDAS, the exponential rule, and the distributive property rules as well as the principle of multiplication of algebraic expressions. In the end, ensure that every term on your expression is in its most simplified form.
How are simplifying expressions and solving equations different?
Solving and simplifying expressions are very different, however, they can be combined the same process because you first need to simplify expressions before you solve them.
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