Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be challenging for budding students in their early years of college or even in high school.
However, learning how to process these equations is essential because it is primary knowledge that will help them move on to higher mathematics and complex problems across various industries.
This article will go over everything you should review to know simplifying expressions. We’ll cover the proponents of simplifying expressions and then test our comprehension through some sample questions.
How Do I Simplify an Expression?
Before you can learn how to simplify them, you must grasp what expressions are in the first place.
In mathematics, expressions are descriptions that have a minimum of two terms. These terms can combine numbers, variables, or both and can be connected through addition or subtraction.
As 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 containing coefficients, variables, and occasionally constants, are also known as polynomials.
Simplifying expressions is essential because it opens up the possibility of grasping how to solve them. Expressions can be expressed in complicated ways, and without simplification, anyone will have a difficult time trying to solve them, with more chance for a mistake.
Of course, each expression differ in how they're simplified based on what terms they contain, but there are common 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, or parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule shows us the order of operations for expressions.
Parentheses. Simplify equations within the parentheses first by adding or subtracting. If there are terms right outside the parentheses, use the distributive property to multiply the term on the outside with the one inside.
Exponents. Where possible, use the exponent rules to simplify the terms that have exponents.
Multiplication and Division. If the equation necessitates it, use multiplication and division to simplify like terms that are applicable.
Addition and subtraction. Finally, add or subtract the resulting terms in the equation.
Rewrite. Ensure that there are no remaining like terms that need to be simplified, and then rewrite the simplified equation.
Here are the Requirements For Simplifying Algebraic Expressions
Beyond the PEMDAS rule, there are a few more rules you need to be aware of when working with algebraic expressions.
You can only apply simplification to 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 variable x as it is.
Parentheses that include another expression directly outside of them need to utilize the distributive property. The distributive property allows you to simplify terms outside of parentheses by distributing them to the terms on the inside, as shown here: a(b+c) = ab + ac.
An extension of the distributive property is known as the property of multiplication. When two separate expressions within parentheses are multiplied, the distributive rule kicks in, and all unique term will will require multiplication by the other terms, resulting in 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 directly outside of an expression in parentheses means that the negative expression must also need to be distributed, changing the signs of the terms on the inside of the parentheses. As is the case in this example: -(8x + 2) will turn into -8x - 2.
Similarly, a plus sign outside the parentheses means that it will have distribution applied to the terms inside. But, this means that you can 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 previous rules were straight-forward enough to use as they only dealt with properties that impact simple terms with variables and numbers. Still, there are additional rules that you need to implement when dealing with exponents and expressions.
In this section, we will review the laws of exponents. 8 properties affect how we utilize exponents, those are the following:
Zero Exponent Rule. This property 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 will not 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 exponents. This is written as am × an = am+n
Quotient Rule. When two terms with matching variables are divided, their quotient subtracts their applicable exponents. This is seen as 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 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 appropriate variables, or (ab)m = am * bm.
Power of a Quotient Rule. In fractional exponents, both the denominator and numerator will take the exponent given, (a/b)m = am/bm.
Simplifying Expressions with the Distributive Property
The distributive property is the rule that denotes that any term multiplied by an expression on the inside of a parentheses needs be multiplied by all of the expressions on the inside. Let’s witness 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.
How to Simplify Expressions with Fractions
Certain expressions can consist of fractions, and just like with exponents, expressions with fractions also have some rules that you need to follow.
When an expression includes 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 denominators and numerators.
Laws of exponents. This states 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 included in the expression. Use the PEMDAS rule and be sure that no two terms share matching variables.
These are the same properties that you can apply when simplifying any real numbers, whether they are decimals, square roots, binomials, logarithms, linear equations, or quadratic 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 PEMDAS and the distributive property. The distributive property will distribute 4 to all other expressions inside of the parentheses, while PEMDAS will decide on the order of simplification.
Because of the distributive property, the term outside of the parentheses will be multiplied by the terms inside.
4(2x) + 4(5x) + 4(7) - 3y
8x + 20x + 28 - 3y
When simplifying equations, remember to add the terms with the same variables, and every term should be in its lowest 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 you should begin with expressions inside parentheses, and in this scenario, that expression also necessitates the distributive property. Here, the term y/4 must be distributed within the two terms on the inside of the parentheses, as seen in this example.
1/3x + y/4(5x) + y/4(2)
Here, let’s put aside the first term for now and simplify the terms with factors attached to them. Since 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 for simplicity 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 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, which tells us that we’ll have to add the exponents of two exponential expressions with like variables multiplied together and multiply their coefficients. This gives us:
24x3 + 4x2 + 18xy + 3y
Due to the fact that there are no more like terms to be simplified, this becomes our final answer.
Simplifying Expressions FAQs
What should I keep in mind when simplifying expressions?
When simplifying algebraic expressions, remember that you must follow the distributive property, PEMDAS, and the exponential rule rules in addition to the rule of multiplication of algebraic expressions. Ultimately, ensure that every term on your expression is in its lowest form.
How are simplifying expressions and solving equations different?
Solving and simplifying expressions are very different, although, they can be incorporated into the same process the same process because you must first simplify expressions before you solve them.
Let Grade Potential Help You Bone Up On Your Math
Simplifying algebraic equations is one of the most foundational precalculus skills you should study. Mastering simplification strategies and laws will pay rewards when you’re practicing advanced mathematics!
But these principles and rules can get complicated fast. But there's no need for you to worry! Grade Potential is here to help!
Grade Potential Las Vegas provides professional teachers that will get you where you need to be at your convenience. Our professional instructors will guide you using mathematical properties in a clear manner to help.
Contact us now!
