Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions are one of the most challenging for new learners in their primary years of college or even in high school.
Still, grasping how to process these equations is important because it is foundational knowledge that will help them eventually be able to solve higher math and complex problems across multiple industries.
This article will discuss everything you should review to master simplifying expressions. We’ll learn the principles of simplifying expressions and then test our skills through some sample questions.
How Do You Simplify Expressions?
Before you can be taught how to simplify expressions, you must understand what expressions are to begin with.
In mathematics, expressions are descriptions that have a minimum of two terms. These terms can combine variables, numbers, or both and can be linked through addition or subtraction.
To give an example, let’s review the following expression.
8x + 2y - 3
This expression includes three terms; 8x, 2y, and 3. The first two contain 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 crucial because it lays the groundwork for learning how to solve them. Expressions can be written in convoluted ways, and without simplification, you will have a tough time trying to solve them, with more chance for solving them incorrectly.
Obviously, all expressions will differ in how they are simplified based on what terms they incorporate, but there are general steps that can be applied to all rational expressions of real numbers, regardless of whether they are logarithms, square roots, etc.
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. Solve equations within the parentheses first by using addition or using subtraction. If there are terms just outside the parentheses, use the distributive property to multiply the term outside with the one inside.
Exponents. Where feasible, use the exponent principles to simplify the terms that have exponents.
Multiplication and Division. If the equation requires it, use multiplication and division to simplify like terms that are applicable.
Addition and subtraction. Lastly, add or subtract the simplified terms in the equation.
Rewrite. Make sure that there are no remaining like terms to simplify, then rewrite the simplified equation.
The Requirements For Simplifying Algebraic Expressions
Beyond the PEMDAS principle, there are a few additional properties you should 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 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 containing 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, as shown here: a(b+c) = ab + ac.
An extension of the distributive property is known as the property of multiplication. When two stand-alone expressions within parentheses are multiplied, the distributive principle is applied, and all separate term will have to be multiplied by the other terms, making 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 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 right outside the parentheses will mean that it will have distribution applied to the terms on the inside. But, 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 alter anything when distributed.
How to Simplify Expressions with Exponents
The previous principles were straight-forward enough to follow as they only applied to properties that affect simple terms with numbers and variables. Despite that, there are a few other rules that you have to implement when working with exponents and expressions.
In this section, we will review the principles of exponents. 8 principles impact how we utilize exponentials, 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 the exponent of 1 won't change in value. Or a1 = a.
Product Rule. When two terms with equivalent variables are multiplied, 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 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 is equivalent to 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 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 have unique variables needs to be applied to the required 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 within parentheses should be multiplied by all of the expressions 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 as with exponents, expressions with fractions also have some rules that you have to follow.
When an expression consist of 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 numerators and denominators.
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 written in the expression. Use the PEMDAS property and be sure that no two terms possess matching variables.
These are the same principles that you can apply when simplifying any real numbers, whether they are decimals, square roots, binomials, quadratic equations, logarithms, or linear equations.
Practice Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this example, the principles that should be noted first are the PEMDAS and the distributive property. The distributive property will distribute 4 to the expressions inside 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 the individual 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 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 first in order should be expressions on the inside of parentheses, and in this scenario, that expression also necessitates the distributive property. Here, the term y/4 should be distributed to 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 the moment and simplify the terms with factors associated with them. Because we know from PEMDAS that fractions 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 because 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 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, meaning 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
Since there are no remaining like terms to simplify, 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 obey PEMDAS, the exponential rule, and the distributive property rules and the principle of multiplication of algebraic expressions. Finally, make sure that every term on your expression is in its most simplified 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 because you first need to simplify expressions before you solve them.
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