Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be scary for beginner students in their first years of high school or college.
However, understanding how to deal with these equations is essential because it is basic information that will help them move on to higher math and complicated problems across various industries.
This article will share everything you must have to know simplifying expressions. We’ll review the principles of simplifying expressions and then test our skills with some practice questions.
How Do You Simplify Expressions?
Before learning how to simplify them, you must learn what expressions are at their core.
In arithmetics, expressions are descriptions that have no less than two terms. These terms can combine variables, numbers, or both and can be connected through addition or subtraction.
For 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 terms include both numbers (8 and 2) and variables (x and y).
Expressions consisting of variables, coefficients, and sometimes constants, are also referred to as polynomials.
Simplifying expressions is important because it paves the way for grasping how to solve them. Expressions can be written in intricate ways, and without simplification, anyone will have a tough time attempting to solve them, with more possibility for a mistake.
Undoubtedly, every expression differ in how they're simplified depending on what terms they include, but there are common steps that can be applied to all rational expressions of real numbers, regardless of whether they are square roots, logarithms, or otherwise.
These steps are called the PEMDAS rule, short for parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule declares the order of operations for expressions.
Parentheses. Simplify equations between the parentheses first by applying addition or applying subtraction. If there are terms right outside the parentheses, use the distributive property to apply multiplication 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 calls for it, utilize the multiplication and division principles to simplify like terms that apply.
Addition and subtraction. Finally, use addition or subtraction the simplified terms in the equation.
Rewrite. Make sure that there are no more like terms that need to be simplified, and rewrite the simplified equation.
Here are the Requirements For Simplifying Algebraic Expressions
In addition to the PEMDAS rule, there are a few more principles you need to be informed of when working 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 leaving the x as it is.
Parentheses that include another expression outside of them need to use the distributive property. The distributive property prompts you to simplify terms on the outside of parentheses by distributing them to the terms inside, for example: a(b+c) = ab + ac.
An extension of the distributive property is called 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. For example: (a + b)(c + d) = a(c + d) + b(c + d).
A negative sign outside an expression in parentheses indicates that the negative expression should also need to be distributed, changing the signs of the terms on the inside of the parentheses. For example: -(8x + 2) will turn into -8x - 2.
Likewise, a plus sign on the outside of 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 change anything when distributed.
How to Simplify Expressions with Exponents
The previous rules were easy enough to use as they only applied to rules that impact simple terms with variables and numbers. Despite that, there are additional rules that you need to implement when dealing with exponents and expressions.
In this section, we will discuss the laws of exponents. 8 principles affect how we deal with exponents, those are the following:
Zero Exponent Rule. This principle states that any term with a 0 exponent is equivalent to 1. Or a0 = 1.
Identity Exponent Rule. Any term with a 1 exponent doesn't change in 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 matching variables are divided by each other, their quotient will subtract their applicable exponents. This is written as 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 that already has an exponent, the term will result in 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 differing variables needs to be applied to the appropriate 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 shows us that any term multiplied by an expression within parentheses needs be multiplied by all of the expressions within. Let’s see 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 resulting expression is 6x + 10.
How to Simplify Expressions with Fractions
Certain expressions contain fractions, and just as with exponents, expressions with fractions also have some rules that you have to follow.
When an expression contains fractions, here is what to remember.
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 tells us that fractions will typically 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 expressed in the expression. Use the PEMDAS principle and make sure that no two terms share matching variables.
These are the same principles that you can apply when simplifying any real numbers, whether they are decimals, square roots, binomials, logarithms, linear equations, or quadratic equations.
Practice Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
Here, the properties that need to 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 on the 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, remember to add the terms with the same variables, and every term should be in its most simplified 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 you should begin with expressions within parentheses, and in this example, that expression also necessitates the distributive property. In this scenario, the term y/4 must be distributed amongst 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 the moment and simplify the terms with factors attached to them. Since we know from PEMDAS that fractions will require multiplication of their denominators and numerators separately, 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 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, 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 other like terms to be simplified, this becomes our final answer.
Simplifying Expressions FAQs
What should I remember when simplifying expressions?
When simplifying algebraic expressions, keep in mind that you are required to obey the exponential rule, the distributive property, and PEMDAS rules as well as the concept of multiplication of algebraic expressions. Ultimately, make sure that every term on your expression is in its lowest form.
How does solving equations differ from simplifying expressions?
Simplifying and solving equations are very different, however, they can be part of the same process the same process since you must first simplify expressions before you solve them.
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