Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be scary for beginner students in their first years of high school or college.
Still, learning how to process these equations is important because it is primary knowledge that will help them move on to higher math and complex problems across different industries.
This article will discuss everything you need to know simplifying expressions. We’ll review the principles of simplifying expressions and then verify our comprehension via some sample problems.
How Does Simplifying Expressions Work?
Before you can learn how to simplify them, you must understand 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 linked through subtraction or addition.
To give an example, let’s go over 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 consisting of coefficients, variables, and occasionally constants, are also known as polynomials.
Simplifying expressions is important because it paves the way for learning how to solve them. Expressions can be written in intricate ways, and without simplification, everyone will have a tough time trying to solve them, with more possibility for error.
Of course, all expressions will be different regarding how they're simplified based on what terms they include, but there are general steps that are applicable to all rational expressions of real numbers, regardless of 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 within the parentheses first by using addition or subtracting. 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 feasible, use the exponent rules to simplify the terms that have exponents.
Multiplication and Division. If the equation necessitates it, utilize the multiplication and division principles 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 additional like terms to simplify, then rewrite the simplified equation.
Here are the Properties For Simplifying Algebraic Expressions
Along with the PEMDAS principle, there are a few more principles you should 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 adding coefficients 8 and 2 and leaving the variable x as it is.
Parentheses that include another expression outside of them need to use the distributive property. The distributive property allows 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 called the principle of multiplication. When two separate expressions within parentheses are multiplied, the distribution property is applied, and all 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 indicates that the negative expression will also need to be distributed, changing the signs of the terms inside 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 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 prior properties were straight-forward enough to implement as they only applied to principles that affect simple terms with numbers and variables. Despite that, there are more rules that you must implement when working with exponents and expressions.
Next, we will review the properties of exponents. Eight principles impact how we deal with exponents, that includes the following:
Zero Exponent Rule. This principle states that any term with the exponent of 0 equals 1. Or a0 = 1.
Identity Exponent Rule. Any term with the exponent of 1 doesn't alter the value. Or a1 = a.
Product Rule. When two terms with the same 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, 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 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 have different variables should be applied to the required 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.
Simplifying Expressions with the Distributive Property
The distributive property is the principle that states that any term multiplied by an expression within parentheses must 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 result is 6x + 10.
How to Simplify Expressions with Fractions
Certain expressions can consist of fractions, and just as with exponents, expressions with fractions also have several rules that you must follow.
When an expression contains fractions, here is what to keep in mind.
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 more likely 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. Refer to the PEMDAS rule and ensure 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, linear equations, quadratic equations, and even logarithms.
Practice Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this case, the rules that should be noted first are the PEMDAS and the distributive property. The distributive property will distribute 4 to all the expressions inside of the parentheses, while PEMDAS will dictate 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, be sure to add the terms with matching variables, and all 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 states that the first in order should be expressions within parentheses, and in this case, that expression also requires the distributive property. In this example, the term y/4 must 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 attached to them. Since we know from PEMDAS that fractions 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 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 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 tells us 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 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, bear in mind that you must obey the distributive property, PEMDAS, and the exponential rule rules and the principle of multiplication of algebraic expressions. Finally, make sure 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 very different, however, they can be incorporated into the same process the same process because you first need to simplify expressions before you solve them.
Let Grade Potential Help You Bone Up On Your Math
Simplifying algebraic equations is a foundational precalculus skills you should practice. Mastering simplification tactics and properties will pay rewards when you’re solving higher mathematics!
But these ideas and rules can get challenging fast. Grade Potential is here to support you, so there is no need to worry!
Grade Potential San Bernardino offers professional instructors that will get you where you need to be at your convenience. Our professional tutors will guide you using mathematical properties in a straight-forward way to assist.
Connect with us now!
