Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be challenging for new students in their primary years of high school or college.
Still, grasping how to deal with these equations is critical because it is foundational knowledge that will help them move on to higher math and advanced problems across different industries.
This article will share everything you must have to master simplifying expressions. We’ll learn the proponents of simplifying expressions and then verify our comprehension via some sample problems.
How Do You Simplify Expressions?
Before you can learn how to simplify expressions, you must learn what expressions are at their core.
In arithmetics, expressions are descriptions that have a minimum of two terms. These terms can contain variables, numbers, or both and can be linked through subtraction or addition.
For example, let’s take a look at the following expression.
8x + 2y - 3
This expression contains three terms; 8x, 2y, and 3. The first two terms contain both numbers (8 and 2) and variables (x and y).
Expressions that include coefficients, variables, and occasionally constants, are also called polynomials.
Simplifying expressions is important because it opens up the possibility of grasping how to solve them. Expressions can be written in intricate ways, and without simplifying them, anyone will have a hard time trying to solve them, with more chance for error.
Of course, each expression vary in how they're simplified based on what terms they include, 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. Solve equations between the parentheses first by adding or applying subtraction. If there are terms right outside the parentheses, use the distributive property to multiply the term on the outside with the one on the inside.
Exponents. Where workable, use the exponent principles to simplify the terms that contain exponents.
Multiplication and Division. If the equation necessitates it, use the multiplication and division principles to simplify like terms that are applicable.
Addition and subtraction. Then, use addition or subtraction the remaining terms in the equation.
Rewrite. Ensure that there are no remaining like terms that need to be simplified, then rewrite the simplified equation.
The Rules For Simplifying Algebraic Expressions
Along with the PEMDAS rule, there are a few more principles you must be aware of when dealing with algebraic expressions.
You can only apply simplification to terms with common variables. When applying addition to these terms, add the coefficient numbers and keep 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 retaining the x as it is.
Parentheses that contain another expression on the 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, for example: a(b+c) = ab + ac.
An extension of the distributive property is known as the concept of multiplication. When two stand-alone expressions within parentheses are multiplied, the distribution property is applied, and each separate term will need to be multiplied by the other terms, making each set of equations, common factors of one another. Like in this example: (a + b)(c + d) = a(c + d) + b(c + d).
A negative sign right outside of an expression in parentheses means that the negative expression must also need to have distribution applied, changing the signs of the terms inside the parentheses. For example: -(8x + 2) will turn into -8x - 2.
Similarly, a plus sign on the outside of the parentheses means that it will have distribution applied to the terms on the inside. Despite that, this means that you can remove the parentheses and write the expression as is owing to the fact that the plus sign doesn’t change anything when distributed.
How to Simplify Expressions with Exponents
The previous principles were simple enough to use as they only dealt with rules that affect simple terms with variables and numbers. Despite that, there are a few other rules that you have to implement when dealing with expressions with exponents.
Here, we will discuss the principles of exponents. 8 rules affect how we utilize exponents, those are the following:
Zero Exponent Rule. This rule states that any term with the exponent of 0 equals 1. Or a0 = 1.
Identity Exponent Rule. Any term with the exponent of 1 won't alter the value. Or a1 = a.
Product Rule. When two terms with equivalent 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 by each other, their quotient will subtract their two respective exponents. This is expressed in 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 result in 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 respective variables, or (ab)m = am * bm.
Power of a Quotient Rule. In fractional exponents, both the numerator and denominator will acquire the exponent given, (a/b)m = am/bm.
How to Simplify Expressions with the Distributive Property
The distributive property is the principle that says that any term multiplied by an expression on the inside of a parentheses must 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.
How to Simplify Expressions with Fractions
Certain expressions can consist of fractions, and just like with exponents, expressions with fractions also have multiple rules that you must 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 shows us that fractions will typically be the power of the quotient rule, which will subtract the exponents of the numerators and denominators.
Simplification. Only fractions at their lowest state should be written in the expression. Apply the PEMDAS rule and make sure that no two terms share the same variables.
These are the same rules that you can apply when simplifying any real numbers, whether they are square roots, binomials, decimals, quadratic equations, logarithms, or linear equations.
Practice Examples for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this example, the properties that must be noted first are the distributive property and the PEMDAS rule. 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 on the 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, remember to add all the terms with the same variables, and each 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 on the inside of parentheses, and in this example, that expression also requires the distributive property. In this example, the term y/4 must be distributed amongst the two terms on the inside of the parentheses, as seen here.
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. Because 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 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 each term to one another, 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 remaining like terms to be simplified, this becomes our final answer.
Simplifying Expressions FAQs
What should I keep in mind when simplifying expressions?
When simplifying algebraic expressions, bear in mind that you must obey the distributive property, PEMDAS, and the exponential rule rules in addition to the rule of multiplication of algebraic expressions. In the end, ensure 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 vastly different, but, they can be incorporated into the same process the same process because you have to simplify expressions before you solve them.
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