Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions are one of the most scary for budding students in their early years of college or even in high school.
Still, grasping how to handle these equations is essential because it is foundational knowledge that will help them eventually be able to solve higher math and complicated problems across multiple industries.
This article will discuss everything you must have to master simplifying expressions. We’ll review the principles of simplifying expressions and then validate our comprehension through some practice problems.
How Do I Simplify an Expression?
Before you can be taught how to simplify them, you must understand what expressions are to begin with.
In arithmetics, expressions are descriptions that have no less than two terms. These terms can contain variables, numbers, or both and can be linked through subtraction or addition.
To give an example, let’s review 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 that include variables, coefficients, and sometimes constants, are also called polynomials.
Simplifying expressions is essential because it lays the groundwork for grasping how to solve them. Expressions can be expressed in complicated ways, and without simplification, anyone will have a tough time attempting to solve them, with more possibility for error.
Undoubtedly, each expression vary 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 known as the PEMDAS rule, short for parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule states that the order of operations for expressions.
Parentheses. Simplify equations within the parentheses first by adding or subtracting. If there are terms just outside the parentheses, use the distributive property to multiply the term on the outside with the one on the inside.
Exponents. Where possible, use the exponent properties to simplify the terms that contain exponents.
Multiplication and Division. If the equation requires it, utilize multiplication or division rules to simplify like terms that apply.
Addition and subtraction. Lastly, add or subtract the resulting terms of the equation.
Rewrite. Make sure that there are no more like terms that need to be simplified, and then rewrite the simplified equation.
The Rules For Simplifying Algebraic Expressions
Along with the PEMDAS sequence, there are a few more rules you need to be aware of when dealing with algebraic expressions.
You can only apply simplification to terms with common variables. When adding these terms, add the coefficient numbers and maintain the variables as [[is|they are]-70. For example, the equation 8x + 2x can be simplified to 10x by applying addition to the coefficients 8 and 2 and leaving the variable x as it is.
Parentheses that include another expression outside of them need to utilize the distributive property. The distributive property gives you the ability to to simplify terms 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 property of multiplication. When two distinct expressions within parentheses are multiplied, the distributive rule kicks in, and all separate term will will require multiplication by the other terms, resulting in each set of equations, common factors of one another. 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 will 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 on the outside of the parentheses means that it will be distributed to the terms on the inside. Despite that, this means that you should remove 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 previous rules were straight-forward enough to follow as they only applied to rules that affect simple terms with numbers and variables. However, there are more rules that you need to follow when dealing with expressions with exponents.
Here, we will discuss the principles of exponents. Eight principles impact how we utilize exponents, which are 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 won't change in value. Or a1 = a.
Product Rule. When two terms with the same variables are apply multiplication, 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 applies subtraction to their respective exponents. This is seen 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 end up having a 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 possess differing variables will be applied to the respective 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 property that states that any term multiplied by an expression within parentheses needs be multiplied by all of the expressions within. 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 expression then becomes 6x + 10.
How to Simplify Expressions with Fractions
Certain expressions can consist of fractions, and just like with exponents, expressions with fractions also have several rules that you must follow.
When an expression includes fractions, here's what to remember.
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 states 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 state should be written in the expression. Use the PEMDAS property and be sure that no two terms contain matching variables.
These are the exact principles that you can apply when simplifying any real numbers, whether they are decimals, square roots, binomials, quadratic equations, logarithms, or linear equations.
Practice Examples 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 distributive property and the PEMDAS rule. The distributive property will distribute 4 to 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 the same variables, and every term should be in its lowest 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 expresses that the the order should start with expressions within parentheses, and in this scenario, that expression also needs the distributive property. In this example, the term y/4 must be distributed to the two terms inside the parentheses, as seen here.
1/3x + y/4(5x) + y/4(2)
Here, let’s put aside the first term for the moment and simplify the terms with factors assigned to them. Remember we know from PEMDAS that fractions require multiplication of their numerators and denominators separately, 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 used 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 the same 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 have to obey the exponential rule, the distributive property, and PEMDAS rules as well as the concept of multiplication of algebraic expressions. Finally, ensure 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 vastly different, although, 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 fundamental precalculus skills you must study. Increasing your skill with simplification strategies and laws will pay dividends when you’re practicing advanced mathematics!
But these concepts and rules can get challenging fast. Grade Potential is here to help, so there is no need to worry!
Grade Potential Pembroke gives professional teachers that will get you where you need to be at your convenience. Our expert tutors will guide you through mathematical principles in a clear way to help.
Book a call now!
