Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions are one of the most challenging for beginner students in their primary years of high school or college.
However, learning how to process these equations is essential because it is basic information that will help them move on to higher arithmetics and complex problems across various industries.
This article will go over everything you must have to master simplifying expressions. We’ll learn the principles of simplifying expressions and then validate our comprehension through some practice questions.
How Do You Simplify Expressions?
Before you can be taught how to simplify them, you must learn what expressions are at their core.
In mathematics, expressions are descriptions that have no less than two terms. These terms can contain numbers, variables, or both and can be linked through subtraction or addition.
To give an example, let’s take a look at the following expression.
8x + 2y - 3
This expression combines three terms; 8x, 2y, and 3. The first two consist of both numbers (8 and 2) and variables (x and y).
Expressions consisting of variables, coefficients, and sometimes constants, are also known as polynomials.
Simplifying expressions is crucial because it opens up the possibility of understanding how to solve them. Expressions can be expressed in intricate ways, and without simplifying them, anyone will have a difficult time attempting to solve them, with more chance for a mistake.
Obviously, every expression differ concerning how they are simplified based on what terms they contain, but there are common steps that apply to all rational expressions of real numbers, whether they are square roots, logarithms, or otherwise.
These steps are known as the PEMDAS rule, an abbreviation for parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule shows us the order of operations for expressions.
Parentheses. Simplify equations within the parentheses first by using addition or applying subtraction. If there are terms right outside the parentheses, use the distributive property to apply multiplication the term 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 calls for it, utilize multiplication or division rules to simplify like terms that are applicable.
Addition and subtraction. Finally, add or subtract the remaining terms of the equation.
Rewrite. Ensure that there are no additional like terms that require simplification, then rewrite the simplified equation.
Here are the Rules For Simplifying Algebraic Expressions
Beyond the PEMDAS rule, there are a few more rules you need to be aware of when dealing 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 expression 8x + 2x can be simplified to 10x by adding coefficients 8 and 2 and retaining 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 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 called the property of multiplication. When two separate expressions within parentheses are multiplied, the distribution property is applied, and all individual 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 outside an expression in parentheses means that the negative expression will also need to have distribution applied, changing the signs of the terms on the inside of the parentheses. Like in this example: -(8x + 2) will turn into -8x - 2.
Likewise, a plus sign right outside the parentheses means that it will have distribution applied to the terms inside. Despite that, this means that you should 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 principles were straight-forward enough to follow as they only applied to principles that impact simple terms with numbers and variables. However, there are additional rules that you need to apply when dealing with expressions with exponents.
Next, we will review the laws of exponents. 8 properties affect how we process exponents, those are the following:
Zero Exponent Rule. This property states that any term with a 0 exponent is equivalent to 1. Or a0 = 1.
Identity Exponent Rule. Any term with a 1 exponent will not alter the value. Or a1 = a.
Product Rule. When two terms with the same 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 will subtract their respective 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 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 that already has 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 possess different variables will be applied to the required 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.
How to Simplify Expressions with the Distributive Property
The distributive property is the property that states 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 witness the distributive property applied 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 multiple rules that you have to follow.
When an expression contains fractions, here's what to keep in mind.
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 usually 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. Use the PEMDAS rule and be sure that no two terms possess the same variables.
These are the same properties that you can apply when simplifying any real numbers, whether they are square roots, binomials, decimals, linear equations, quadratic equations, and even logarithms.
Sample Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this case, the rules 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.
Because of the distributive property, the term outside of 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 the terms with matching variables, and every term should be in its most simplified form.
28x + 28 - 3y
Rearrange the equation like this:
28x - 3y + 28
Example 2
Simplify the expression 1/3x + y/4(5x + 2)
The PEMDAS rule expresses that the you should begin with expressions within parentheses, and in this scenario, that expression also requires the distributive property. In this scenario, the term y/4 must be distributed within 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. Since we know from PEMDAS that fractions will require multiplication of their denominators and numerators individually, we will then have:
y/4 * 5x/1
The expression 5x/1 is used to keep things simple since 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 utilized to distribute all terms 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 the same 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 be simplified, this becomes our final answer.
Simplifying Expressions FAQs
What should I bear in mind when simplifying expressions?
When simplifying algebraic expressions, bear in mind that you have to follow the exponential rule, the distributive property, and PEMDAS rules in addition to the concept of multiplication of algebraic expressions. Finally, make sure that every term on your expression is in its lowest form.
What is the difference between solving an equation and simplifying an expression?
Solving equations and simplifying expressions are quite different, although, they can be incorporated into the same process the same process because you must first simplify expressions before you begin solving them.
Let Grade Potential Help You Hone Your Math Skills
Simplifying algebraic equations is a fundamental precalculus skills you need to study. Getting proficient at simplification strategies and rules will pay dividends when you’re practicing higher mathematics!
But these concepts and rules can get challenging fast. But there's no need for you to worry! Grade Potential is here to support you!
Grade Potential Clear Water offers expert teachers that will get you where you need to be at your convenience. Our experienced tutors will guide you applying mathematical concepts in a step-by-step manner to assist.
Book a call now!
