Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can appear to be scary for budding learners in their early years of college or even in high school.
However, learning how to deal with these equations is essential because it is primary knowledge that will help them navigate higher math and complicated problems across multiple industries.
This article will share everything you must have to know simplifying expressions. We’ll review the proponents of simplifying expressions and then validate our skills with some practice problems.
How Do I Simplify an Expression?
Before learning how to simplify expressions, you must grasp what expressions are to begin with.
In arithmetics, expressions are descriptions that have at least two terms. These terms can contain variables, numbers, or both and can be connected through subtraction or addition.
For example, let’s go over the following expression.
8x + 2y - 3
This expression contains three terms; 8x, 2y, and 3. The first two terms include both numbers (8 and 2) and variables (x and y).
Expressions that incorporate variables, coefficients, and sometimes constants, are also called polynomials.
Simplifying expressions is essential because it lays the groundwork for learning how to solve them. Expressions can be written in intricate ways, and without simplifying them, you will have a hard time trying to solve them, with more opportunity for a mistake.
Undoubtedly, each expression differ concerning how they are simplified based on what terms they include, but there are common steps that are applicable to all rational expressions of real numbers, regardless of whether they are square roots, logarithms, or otherwise.
These steps are refered to as the PEMDAS rule, or parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule declares the order of operations for expressions.
Parentheses. Solve equations within the parentheses first by using addition or applying subtraction. If there are terms just outside the parentheses, use the distributive property to multiply the term on the outside with the one inside.
Exponents. Where possible, use the exponent principles to simplify the terms that contain exponents.
Multiplication and Division. If the equation necessitates it, utilize multiplication or division rules to simplify like terms that apply.
Addition and subtraction. Then, use addition or subtraction the simplified terms in the equation.
Rewrite. Ensure that there are no remaining like terms that need to be simplified, and then rewrite the simplified equation.
Here are the Rules For Simplifying Algebraic Expressions
In addition to the PEMDAS rule, there are a few more rules you should be aware of when working with algebraic expressions.
You can only simplify terms with common variables. When applying addition to these terms, add the coefficient numbers and maintain the variables as [[is|they are]-70. For example, the equation 8x + 2x can be simplified to 10x by adding coefficients 8 and 2 and keeping the variable x as it is.
Parentheses that include another expression directly outside of them need to utilize the distributive property. The distributive property gives you the ability to to simplify terms on the outside of parentheses by distributing them to the terms on the inside, or as follows: a(b+c) = ab + ac.
An extension of the distributive property is referred to as the principle of multiplication. When two separate expressions within parentheses are multiplied, the distributive principle applies, and all individual term will need to be multiplied by the other terms, resulting in each set of equations, common factors of one another. For example: (a + b)(c + d) = a(c + d) + b(c + d).
A negative sign right outside of an expression in parentheses denotes that the negative expression will also need to have distribution applied, changing the signs of the terms inside the parentheses. Like in this example: -(8x + 2) will turn into -8x - 2.
Similarly, a plus sign on the outside of the parentheses denotes 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 rules were easy enough to use as they only applied to rules that impact simple terms with numbers and variables. Despite that, there are more rules that you need to apply when working 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 equal to 1. Or a0 = 1.
Identity Exponent Rule. Any term with the exponent of 1 doesn't change in value. Or a1 = a.
Product Rule. When two terms with equivalent variables are multiplied, their product will add their exponents. This is written as am × an = am+n
Quotient Rule. When two terms with the same variables are divided, their quotient subtracts their two respective exponents. This is seen as the formula am/an = am-n.
Negative Exponents Rule. Any term with a negative exponent is equivalent to 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 being the product of the two exponents that were applied to it, or (am)n = amn.
Power of a Product Rule. An exponent applied to two terms that have different variables should be applied to the required variables, or (ab)m = am * bm.
Power of a Quotient Rule. In fractional exponents, both the numerator and denominator will assume the exponent given, (a/b)m = am/bm.
How to Simplify Expressions with the Distributive Property
The distributive property is the property that says that any term multiplied by an expression on the inside of a parentheses needs be multiplied by all of the expressions on the inside. Let’s watch the distributive property in action 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 expression then becomes 6x + 10.
Simplifying Expressions with Fractions
Certain expressions contain fractions, and just as with exponents, expressions with fractions also have several rules that you must follow.
When an expression has 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 numerators and denominators.
Laws of exponents. This tells us that fractions will more likely be the power of the quotient rule, which will apply subtraction to the exponents of the denominators and numerators.
Simplification. Only fractions at their lowest form should be written in the expression. Refer to the PEMDAS rule and be sure that no two terms possess matching variables.
These are the exact properties that you can apply when simplifying any real numbers, whether they are decimals, square roots, binomials, linear equations, quadratic equations, and even logarithms.
Sample Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this example, the principles that should be noted first are the distributive property and the PEMDAS rule. The distributive property will distribute 4 to all the expressions inside the parentheses, while PEMDAS will dictate the order of simplification.
Due to the distributive property, the term outside of the parentheses will be multiplied by each term on the inside.
4(2x) + 4(5x) + 4(7) - 3y
8x + 20x + 28 - 3y
When simplifying equations, remember to add all the terms with the same variables, and each 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 states that the the order should start with expressions inside parentheses, and in this example, that expression also requires the distributive property. Here, the term y/4 must be distributed to the two terms inside the parentheses, as seen in this example.
1/3x + y/4(5x) + y/4(2)
Here, let’s set aside the first term for the moment and simplify the terms with factors attached to them. Since we know from PEMDAS that fractions will require multiplication of their numerators and denominators separately, we will then have:
y/4 * 5x/1
The expression 5x/1 is used for simplicity because 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 each 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, meaning that we’ll have to add the exponents of two exponential expressions with like variables multiplied together and multiply their coefficients. This gives us:
24x3 + 4x2 + 18xy + 3y
Since there are no more like terms to simplify, this becomes our final answer.
Simplifying Expressions FAQs
What should I keep in mind when simplifying expressions?
When simplifying algebraic expressions, remember that you are required to follow the exponential rule, the distributive property, and PEMDAS rules as well as the principle of multiplication of algebraic expressions. Ultimately, ensure that every term on your expression is in its most simplified form.
How does solving equations differ from simplifying expressions?
Solving equations and simplifying expressions are quite different, however, they can be part of the same process the same process since you have to simplify expressions before solving them.
Let Grade Potential Help You Get a Grip on Math
Simplifying algebraic equations is one of the most foundational precalculus skills you need to practice. Increasing your skill with simplification tactics and laws will pay dividends when you’re learning higher mathematics!
But these ideas and properties can get complicated fast. Have no fear though! Grade Potential is here to help!
Grade Potential Oakland offers expert tutors that will get you where you need to be at your convenience. Our expert teachers will guide you through mathematical concepts in a clear manner to guide.
Book a call now!
