Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be scary for beginner pupils in their early years of college or even in high school.
Nevertheless, grasping how to process these equations is important because it is primary information that will help them move on to higher mathematics and complicated problems across multiple industries.
This article will share everything you need to know simplifying expressions. We’ll review the principles of simplifying expressions and then verify our skills via some sample questions.
How Do You Simplify Expressions?
Before you can be taught 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 contain variables, numbers, or both and can be linked through subtraction or addition.
To give an example, let’s review the following expression.
8x + 2y - 3
This expression includes three terms; 8x, 2y, and 3. The first two include both numbers (8 and 2) and variables (x and y).
Expressions consisting of coefficients, variables, and occasionally constants, are also referred to as polynomials.
Simplifying expressions is crucial because it paves the way for learning how to solve them. Expressions can be written in convoluted ways, and without simplifying them, anyone will have a hard time attempting to solve them, with more possibility for a mistake.
Undoubtedly, each expression vary concerning how they're simplified based on what terms they incorporate, but there are common steps that are applicable to all rational expressions of real numbers, regardless of whether they are logarithms, square roots, etc.
These steps are called the PEMDAS rule, an abbreviation for parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule states that the order of operations for expressions.
Parentheses. Solve equations between the parentheses first by adding or subtracting. If there are terms just outside the parentheses, use the distributive property to multiply the term outside with the one inside.
Exponents. Where possible, use the exponent properties to simplify the terms that contain exponents.
Multiplication and Division. If the equation necessitates it, use multiplication or division rules to simplify like terms that are applicable.
Addition and subtraction. Finally, add or subtract the remaining terms in the equation.
Rewrite. Make sure that there are no remaining like terms that require simplification, and then rewrite the simplified equation.
The Properties For Simplifying Algebraic Expressions
Beyond the PEMDAS principle, there are a few additional rules you need to be aware of when simplifying 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 leaving the variable x as it is.
Parentheses that contain another expression on the outside of them need to apply the distributive property. The distributive property allows you to simplify terms outside of parentheses by distributing them to the terms inside, or as follows: a(b+c) = ab + ac.
An extension of the distributive property is referred to as the property of multiplication. When two stand-alone expressions within parentheses are multiplied, the distribution rule kicks in, and all separate term will have to be multiplied by the other terms, resulting in each set of equations, common factors of one another. Such as is the case here: (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 will also need to have distribution applied, changing the signs of the terms on the inside of the parentheses. As is the case in this example: -(8x + 2) will turn into -8x - 2.
Likewise, a plus sign on the outside of the parentheses means that it will be distributed to the terms on the inside. But, this means that you can eliminate the parentheses and write the expression as is because the plus sign doesn’t alter anything when distributed.
How to Simplify Expressions with Exponents
The prior rules were simple enough to follow as they only applied to properties that affect simple terms with variables and numbers. Still, there are more rules that you need to implement when dealing with expressions with exponents.
Next, we will talk about the properties of exponents. Eight principles impact how we deal with exponentials, that includes the following:
Zero Exponent Rule. This principle states that any term with the exponent of 0 is equal to 1. Or a0 = 1.
Identity Exponent Rule. Any term with the exponent of 1 will not alter the value. Or a1 = a.
Product Rule. When two terms with the same variables are multiplied, their product will add their two exponents. This is expressed in the formula am × an = am+n
Quotient Rule. When two terms with the same variables are divided, their quotient subtracts their applicable exponents. This is written as the formula am/an = am-n.
Negative Exponents Rule. Any term with a negative exponent is equal to the inverse of that term over 1. This is expressed with 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 applied to it, or (am)n = amn.
Power of a Product Rule. An exponent applied to two terms that possess different variables will 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 acquire the exponent given, (a/b)m = am/bm.
Simplifying 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 see 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 resulting expression is 6x + 10.
Simplifying Expressions with Fractions
Certain expressions can consist of fractions, and just like with exponents, expressions with fractions also have some rules that you have to follow.
When an expression consist of fractions, here is what to keep in mind.
Distributive property. The distributive property a(b+c) = ab + ac, when applied to fractions, will multiply fractions one at a time by their denominators and numerators.
Laws of exponents. This states that fractions will usually be the power of the quotient rule, which will subtract the exponents of the denominators and numerators.
Simplification. Only fractions at their lowest should be written in the expression. Use the PEMDAS principle and ensure that no two terms share the same variables.
These are the same properties that you can apply when simplifying any real numbers, whether they are binomials, decimals, square roots, quadratic equations, logarithms, or linear equations.
Practice Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this example, the rules that need to be noted first are PEMDAS and the distributive property. The distributive property will distribute 4 to all other expressions inside of the parentheses, while PEMDAS will dictate the order of simplification.
As a result of the distributive property, the term outside the parentheses will be multiplied by the terms inside.
4(2x) + 4(5x) + 4(7) - 3y
8x + 20x + 28 - 3y
When simplifying equations, be sure 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 first in order should be expressions within parentheses, and in this scenario, that expression also necessitates the distributive property. Here, the term y/4 must be distributed to the two terms within the parentheses, as follows.
1/3x + y/4(5x) + y/4(2)
Here, let’s set aside the first term for now and simplify the terms with factors attached to them. Remember 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 to keep things simple 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 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, which tells us 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
Due to the fact that there are no remaining like terms to apply simplification to, this becomes our final answer.
Simplifying Expressions FAQs
What should I remember when simplifying expressions?
When simplifying algebraic expressions, bear in mind that you have to obey the distributive property, PEMDAS, and the exponential rule rules as well as the principle of multiplication of algebraic expressions. Ultimately, ensure that every term on your expression is in its lowest form.
How does solving equations differ from simplifying expressions?
Simplifying and solving equations are vastly different, however, they can be combined the same process because you first need to simplify expressions before you solve them.
Let Grade Potential Help You Get a Grip on Math
Simplifying algebraic equations is a primary precalculus skills you need to study. Increasing your skill with simplification strategies and properties will pay rewards when you’re learning sophisticated mathematics!
But these principles and rules can get complicated fast. Grade Potential is here to support you, so have no fear!
Grade Potential Seattle gives professional teachers that will get you on top of your skills at your convenience. Our experienced instructors will guide you using mathematical concepts in a clear way to guide.
Book a call now!
