Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions are one of the most scary for beginner pupils in their early years of college or even in high school.
Nevertheless, understanding how to deal with these equations is critical because it is foundational knowledge that will help them eventually be able to solve higher mathematics and complicated problems across various industries.
This article will go over everything you should review to master simplifying expressions. We’ll learn the proponents of simplifying expressions and then verify our skills through some practice problems.
How Do You Simplify Expressions?
Before you can learn how to simplify them, you must learn what expressions are to begin with.
In mathematics, expressions are descriptions that have a minimum of two terms. These terms can combine numbers, variables, or both and can be linked through addition or subtraction.
For example, let’s review the following expression.
8x + 2y - 3
This expression includes three terms; 8x, 2y, and 3. The first two terms include both numbers (8 and 2) and variables (x and y).
Expressions that include variables, coefficients, and sometimes constants, are also called polynomials.
Simplifying expressions is essential because it paves the way for understanding how to solve them. Expressions can be written in intricate ways, and without simplifying them, you will have a tough time trying to solve them, with more possibility for solving them incorrectly.
Obviously, every expression be different regarding how they're simplified depending on what terms they include, but there are typical steps that apply 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, an abbreviation for parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule states that the order of operations for expressions.
Parentheses. Resolve equations within the parentheses first by using addition or using 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 workable, use the exponent rules to simplify the terms that have exponents.
Multiplication and Division. If the equation calls for it, use multiplication and division to simplify like terms that are applicable.
Addition and subtraction. Then, use addition or subtraction the simplified terms of the equation.
Rewrite. Ensure that there are no remaining like terms to simplify, and rewrite the simplified equation.
Here are the Rules For Simplifying Algebraic Expressions
In addition to the PEMDAS rule, there are a few additional principles you need to be informed of when simplifying algebraic expressions.
You can only apply simplification to terms with common variables. When applying addition to these terms, add the coefficient numbers and leave 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 variable x as it is.
Parentheses that include another expression outside of them need to apply 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 concept of multiplication. When two stand-alone expressions within parentheses are multiplied, the distributive rule is applied, and every individual term will have to be multiplied by the other terms, resulting in each set of equations, common factors of each other. Like in this example: (a + b)(c + d) = a(c + d) + b(c + d).
A negative sign outside an expression in parentheses denotes that the negative expression should also need to be distributed, 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 on the inside. Despite that, this means that you are able to eliminate the parentheses and write the expression as is due to the fact that the plus sign doesn’t change anything when distributed.
How to Simplify Expressions with Exponents
The prior rules were straight-forward enough to follow as they only applied to rules that impact simple terms with numbers and variables. However, there are more rules that you must follow when working with expressions with exponents.
Next, we will talk about the properties of exponents. Eight properties influence how we deal with exponents, which are the following:
Zero Exponent Rule. This property states that any term with a 0 exponent equals 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 apply multiplication, 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, their quotient applies subtraction to their two 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 already with an exponent, the term will end up having a 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 appropriate 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 rule that says that any term multiplied by an expression within parentheses must be multiplied by all of the expressions within. Let’s watch 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 result 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 need 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 shows 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 expressed in the expression. Apply the PEMDAS rule and ensure that no two terms share matching variables.
These are the same principles that you can apply when simplifying any real numbers, whether they are square roots, binomials, decimals, quadratic equations, logarithms, or linear 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 PEMDAS and the distributive property. The distributive property will distribute 4 to all the expressions inside of the parentheses, while PEMDAS will govern the order of simplification.
Because 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, you should add the terms with matching variables, and each term should be in its most simplified form.
28x + 28 - 3y
Rearrange the equation this way:
28x - 3y + 28
Example 2
Simplify the expression 1/3x + y/4(5x + 2)
The PEMDAS rule states that the you should begin with expressions on the inside of parentheses, and in this scenario, that expression also requires the distributive property. In this scenario, the term y/4 will need to be distributed within the two terms within 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 assigned to them. Since we know from PEMDAS that fractions will require multiplication of their numerators and denominators individually, we will then have:
y/4 * 5x/1
The expression 5x/1 is used for simplicity 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 utilized 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 the same variables multiplied together and multiply their coefficients. This gives us:
24x3 + 4x2 + 18xy + 3y
Since there are no remaining like terms to be simplified, this becomes our final answer.
Simplifying Expressions FAQs
What should I remember when simplifying expressions?
When simplifying algebraic expressions, bear in mind that you are required to follow the distributive property, PEMDAS, and the exponential rule rules and the rule of multiplication of algebraic expressions. Finally, make sure that every term on your expression is in its lowest form.
How are simplifying expressions and solving equations different?
Solving equations and simplifying expressions are very different, however, they can be part of the same process the same process because you have to simplify expressions before you begin solving them.
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