Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions are one of the most intimidating for new pupils in their early years of high school or college.
Nevertheless, learning how to process these equations is critical because it is basic knowledge that will help them move on to higher mathematics and complicated problems across multiple industries.
This article will share everything you should review to know simplifying expressions. We’ll cover the proponents of simplifying expressions and then test our comprehension via some practice problems.
How Do I Simplify an Expression?
Before learning how to simplify expressions, you must grasp what expressions are in the first place.
In arithmetics, expressions are descriptions that have a minimum of two terms. These terms can contain variables, numbers, or both and can be connected through subtraction or addition.
As an example, let’s go over the following expression.
8x + 2y - 3
This expression includes 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 essential because it lays the groundwork for grasping how to solve them. Expressions can be written in convoluted ways, and without simplifying them, anyone will have a hard time attempting to solve them, with more chance for a mistake.
Of course, every expression differ regarding how they are simplified based on what terms they incorporate, but there are typical 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 called the PEMDAS rule, or parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule shows us the order of operations for expressions.
Parentheses. Resolve equations between the parentheses first by adding or applying subtraction. If there are terms right outside the parentheses, use the distributive property to apply multiplication the term on the outside with the one inside.
Exponents. Where possible, use the exponent properties 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. Finally, use addition or subtraction the remaining terms in the equation.
Rewrite. Ensure that there are no more like terms to simplify, and rewrite the simplified equation.
Here are the Rules For Simplifying Algebraic Expressions
Beyond the PEMDAS principle, there are a few more properties you must be informed of when working with algebraic expressions.
You can only simplify terms with common variables. When applying addition to these terms, add the coefficient numbers and maintain 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 retaining the x as it is.
Parentheses that include another expression outside of them need to apply the distributive property. The distributive property gives you the ability to 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 separate expressions within parentheses are multiplied, the distributive principle kicks in, and every unique term will have to be multiplied 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 outside an expression in parentheses means that the negative expression should also need to be distributed, changing the signs of the terms inside the parentheses. Like in this example: -(8x + 2) will turn into -8x - 2.
Likewise, a plus sign outside the parentheses denotes that it will have distribution applied to the terms on the inside. But, this means that you can 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 rules were easy enough to use as they only applied to rules that impact simple terms with numbers and variables. Still, there are additional rules that you need to implement when working with expressions with exponents.
Here, we will discuss the principles of exponents. 8 principles affect how we utilize exponents, that includes the following:
Zero Exponent Rule. This principle states that any term with a 0 exponent 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 equivalent 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 by each other, their quotient applies subtraction to their applicable 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 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 that were applied to it, or (am)n = amn.
Power of a Product Rule. An exponent applied to two terms that possess differing 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 assume 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 should be multiplied by all of the expressions 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 expression then becomes 6x + 10.
How to Simplify Expressions with Fractions
Certain expressions contain fractions, and just like with exponents, expressions with fractions also have some rules that you must follow.
When an expression includes 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 denominators and numerators.
Laws of exponents. This shows 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 should be included in the expression. Use the PEMDAS property and be sure that no two terms contain matching 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 Examples for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
Here, the rules that should be noted first are the distributive property and the PEMDAS rule. The distributive property will distribute 4 to all the expressions on the inside of the parentheses, while PEMDAS will decide on the order of simplification.
Due to the distributive property, the term outside of 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, you should 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 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 to 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 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 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 used 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, 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
Because there are no other 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, keep in mind that you are required to follow the exponential rule, the distributive property, and PEMDAS rules and the principle 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?
Solving and simplifying expressions are vastly different, although, they can be incorporated into the same process the same process due to the fact that you must first simplify expressions before you begin solving them.
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