Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be challenging for beginner pupils in their primary years of high school or college.
Still, understanding how to handle these equations is critical because it is basic information that will help them move on to higher arithmetics and complex problems across various industries.
This article will discuss everything you need to master simplifying expressions. We’ll review the laws of simplifying expressions and then test what we've learned through some practice questions.
How Does Simplifying Expressions Work?
Before you can learn how to simplify them, you must learn what expressions are at their core.
In arithmetics, expressions are descriptions that have no less than two terms. These terms can include numbers, variables, or both and can be linked through addition or subtraction.
As an example, let’s take a look at the following expression.
8x + 2y - 3
This expression combines three terms; 8x, 2y, and 3. The first two terms contain both numbers (8 and 2) and variables (x and y).
Expressions that incorporate coefficients, variables, and occasionally constants, are also referred to as polynomials.
Simplifying expressions is essential because it paves the way for understanding how to solve them. Expressions can be written in intricate ways, and without simplification, anyone will have a tough time attempting to solve them, with more chance for solving them incorrectly.
Of course, each expression vary concerning how they're simplified based on what terms they incorporate, but there are general steps that apply to all rational expressions of real numbers, regardless of whether they are logarithms, square roots, etc.
These steps are called the PEMDAS rule, an abbreviation for parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule declares the order of operations for expressions.
Parentheses. Solve equations within the parentheses first by adding or applying subtraction. If there are terms just outside the parentheses, use the distributive property to apply multiplication the term on the outside with the one on the inside.
Exponents. Where feasible, use the exponent principles to simplify the terms that include exponents.
Multiplication and Division. If the equation calls for it, use multiplication and division to simplify like terms that apply.
Addition and subtraction. Lastly, add or subtract the remaining terms of the equation.
Rewrite. Make sure that there are no more like terms to simplify, then rewrite the simplified equation.
Here are the Rules For Simplifying Algebraic Expressions
In addition to the PEMDAS rule, there are a few additional principles you should be aware of when dealing with algebraic expressions.
You can only simplify terms with common variables. When adding these terms, add the coefficient numbers and leave 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 x as it is.
Parentheses that contain another expression directly outside of them need to utilize 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 called the property of multiplication. When two separate expressions within parentheses are multiplied, the distributive property kicks in, and all separate term will need 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 right outside of an expression in parentheses indicates that the negative expression should also need to have distribution applied, changing the signs of the terms inside the parentheses. As is the case in this example: -(8x + 2) will turn into -8x - 2.
Likewise, a plus sign on the outside of the parentheses means that it will have distribution applied to the terms inside. But, this means that you are able to remove 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 prior properties were straight-forward enough to use as they only dealt with rules that impact simple terms with variables and numbers. Despite that, there are more rules that you must implement when dealing with exponents and expressions.
In this section, we will talk about the laws of exponents. 8 principles influence how we utilize exponentials, those are the following:
Zero Exponent Rule. This principle states that any term with the exponent of 0 is equivalent to 1. Or a0 = 1.
Identity Exponent Rule. Any term with a 1 exponent doesn't change in value. Or a1 = a.
Product Rule. When two terms with equivalent variables are apply multiplication, their product will add their exponents. This is written as am × an = am+n
Quotient Rule. When two terms with the same variables are divided, their quotient subtracts their two 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 unique variables will be applied to the respective 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 rule that shows us that any term multiplied by an expression on the inside of a parentheses should 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.
Simplifying Expressions with Fractions
Certain expressions contain fractions, and just like with exponents, expressions with fractions also have several 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 denominators and numerators.
Laws of exponents. This tells us that fractions will usually be the power of the quotient rule, which will apply subtraction to the exponents of the numerators and denominators.
Simplification. Only fractions at their lowest form should be included in the expression. Use the PEMDAS property and be sure that no two terms contain the same variables.
These are the same properties that you can apply when simplifying any real numbers, whether they are square roots, binomials, decimals, quadratic equations, logarithms, or linear equations.
Practice Examples for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this example, the properties that should be noted first are the distributive property and the PEMDAS rule. The distributive property will distribute 4 to all the expressions inside the parentheses, while PEMDAS will dictate the order of simplification.
Due to 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 all the terms with the same variables, and all 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 the order should start with expressions on the inside of parentheses, and in this case, that expression also requires 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 now and simplify the terms with factors attached to them. Remember we know from PEMDAS that fractions will require multiplication of their numerators and denominators individually, 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 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, which tells us 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 more like terms to simplify, 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 must obey 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.
What is the difference between solving an equation and simplifying an expression?
Solving equations and simplifying expressions are quite different, however, 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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