Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be intimidating for new learners in their early years of high school or college.
Nevertheless, learning how to handle these equations is important because it is foundational knowledge that will help them navigate higher mathematics and complicated problems across multiple industries.
This article will go over everything you must have to know simplifying expressions. We’ll learn the proponents of simplifying expressions and then validate what we've learned with some sample problems.
How Do I Simplify an Expression?
Before learning how to simplify expressions, you must grasp what expressions are at their core.
In arithmetics, expressions are descriptions that have a minimum of two terms. These terms can contain numbers, variables, or both and can be connected through addition or subtraction.
To give an example, let’s go over 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 containing 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 convoluted ways, and without simplification, everyone will have a difficult time trying to solve them, with more possibility for a mistake.
Undoubtedly, all expressions will differ in how they are simplified based on what terms they include, but there are general 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 refered to 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 inside the parentheses first by adding or using subtraction. If there are terms right outside the parentheses, use the distributive property to multiply the term outside with the one inside.
Exponents. Where workable, use the exponent principles to simplify the terms that include exponents.
Multiplication and Division. If the equation calls for it, utilize multiplication and division to simplify like terms that are applicable.
Addition and subtraction. Lastly, add or subtract the resulting terms of the equation.
Rewrite. Make sure that there are no more like terms to simplify, and then rewrite the simplified equation.
The Properties For Simplifying Algebraic Expressions
Beyond the PEMDAS sequence, there are a few more rules you must be aware of when working with 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 applying addition to the coefficients 8 and 2 and keeping the variable x as it is.
Parentheses containing another expression directly outside of them need to apply the distributive property. The distributive property prompts 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 distributive principle is applied, and every unique term will will require multiplication by the other terms, making 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 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 right outside the parentheses will mean that it will be distributed to the terms inside. Despite that, this means that you should 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 previous properties were simple enough to follow as they only applied to properties that impact simple terms with numbers and variables. However, there are more rules that you have to apply when dealing with expressions with exponents.
Here, we will talk about the laws of exponents. 8 principles impact how we process exponents, those are the following:
Zero Exponent Rule. This principle states that any term with the exponent of 0 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 equivalent 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, their quotient will subtract their applicable exponents. This is expressed in 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 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 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 have differing variables needs to 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.
Simplifying Expressions with the Distributive Property
The distributive property is the principle that shows us 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 watch the distributive property used 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 contain fractions, and just as with exponents, expressions with fractions also have multiple rules that you need to follow.
When an expression has fractions, here's 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 numerators and denominators.
Laws of exponents. This states that fractions will typically be the power of the quotient rule, which will subtract the exponents of the denominators and numerators.
Simplification. Only fractions at their lowest form should be written in the expression. Refer to the PEMDAS rule and be sure that no two terms share the same variables.
These are the exact rules 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 case, the rules that need to be noted first are the PEMDAS and the distributive property. The distributive property will distribute 4 to the expressions inside of the parentheses, while PEMDAS will decide on the order of simplification.
Because of the distributive property, the term outside the parentheses will be multiplied by each term on the 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 this way:
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. Here, the term y/4 should be distributed to the two terms within the parentheses, as seen here.
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 associated with them. Remember 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 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 means that we’ll have to add the exponents of two exponential expressions with similar variables multiplied together and multiply their coefficients. This gives us:
24x3 + 4x2 + 18xy + 3y
Due to the fact that there are no other 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, remember that you must follow the distributive property, PEMDAS, and the exponential rule rules as well as the rule of multiplication of algebraic expressions. Finally, ensure that every term on your expression is in its most simplified form.
How are simplifying expressions and solving equations different?
Simplifying and solving equations are vastly different, but, 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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