Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions are one of the most intimidating for beginner students in their first years of high school or college.
Nevertheless, understanding how to handle these equations is critical because it is foundational information that will help them navigate higher math and advanced problems across various industries.
This article will share everything you need to learn simplifying expressions. We’ll review the laws of simplifying expressions and then verify our skills with some practice problems.
How Does Simplifying Expressions Work?
Before you can be taught how to simplify expressions, you must understand what expressions are to begin with.
In mathematics, expressions are descriptions that have a minimum of two terms. These terms can combine numbers, variables, or both and can be linked through addition or subtraction.
As an example, let’s go over the following expression.
8x + 2y - 3
This expression combines three terms; 8x, 2y, and 3. The first two include both numbers (8 and 2) and variables (x and y).
Expressions containing variables, coefficients, and sometimes constants, are also referred to as polynomials.
Simplifying expressions is crucial because it lays the groundwork for learning how to solve them. Expressions can be expressed in intricate ways, and without simplification, anyone will have a tough time trying to solve them, with more possibility for a mistake.
Of course, each expression vary regarding how they are 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 square roots, logarithms, or otherwise.
These steps are known as the PEMDAS rule, or parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule states that the order of operations for expressions.
Parentheses. Resolve equations inside the parentheses first by applying addition or using subtraction. 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 principles to simplify the terms that contain exponents.
Multiplication and Division. If the equation requires it, use multiplication and division to simplify like terms that are applicable.
Addition and subtraction. Finally, add or subtract the simplified terms in the equation.
Rewrite. Make sure that there are no additional like terms that require simplification, and then rewrite the simplified equation.
Here are the Requirements For Simplifying Algebraic Expressions
Beyond the PEMDAS sequence, there are a few more rules you must be aware of when dealing with algebraic expressions.
You can only apply simplification to terms with common variables. When applying addition to 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 variable x as it is.
Parentheses containing another expression directly outside of them need to apply the distributive property. The distributive property prompts you to simplify terms outside of parentheses by distributing them to the terms on the inside, for example: a(b+c) = ab + ac.
An extension of the distributive property is known as the property of multiplication. When two stand-alone expressions within parentheses are multiplied, the distributive rule kicks in, and each unique term will have to be multiplied by the other terms, making 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 right outside of an expression in parentheses indicates that the negative expression must also need to be distributed, changing the signs of the terms on the inside of the parentheses. Like in this example: -(8x + 2) will turn into -8x - 2.
Similarly, a plus sign outside the parentheses will mean that it will have distribution applied to the terms on the inside. However, 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 prior rules were straight-forward enough to use as they only applied to properties that affect simple terms with numbers and variables. However, there are additional rules that you must implement when working with expressions with exponents.
Next, we will discuss the principles of exponents. 8 principles affect how we process exponentials, which are the following:
Zero Exponent Rule. This principle states that any term with a 0 exponent is equivalent to 1. Or a0 = 1.
Identity Exponent Rule. Any term with a 1 exponent doesn't alter the value. Or a1 = a.
Product Rule. When two terms with equivalent 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 the same 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 equal 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 end up being the product of the two exponents applied to it, or (am)n = amn.
Power of a Product Rule. An exponent applied to two terms that have different variables will 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 assume the exponent given, (a/b)m = am/bm.
Simplifying Expressions with the Distributive Property
The distributive property is the principle that denotes that any term multiplied by an expression on the inside of a parentheses must be multiplied by all of the expressions within. Let’s see 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.
Simplifying Expressions with Fractions
Certain expressions contain fractions, and just as with exponents, expressions with fractions also have multiple 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 numerators and denominators.
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 expressed in the expression. Use the PEMDAS property and be sure that no two terms share matching variables.
These are the same principles that you can apply when simplifying any real numbers, whether they are decimals, square roots, binomials, logarithms, linear equations, or quadratic equations.
Practice Examples for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
Here, the properties that should be noted first are the distributive property and the PEMDAS rule. The distributive property will distribute 4 to all other expressions inside of the parentheses, while PEMDAS will govern the order of simplification.
As a result of 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, be sure to 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 the order should start with expressions inside parentheses, and in this scenario, that expression also requires the distributive property. Here, the term y/4 will need to be distributed to the two terms within 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 assigned to them. Since we know from PEMDAS that fractions will need to multiply 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 utilized to distribute every term 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 remaining like terms to apply simplification to, this becomes our final answer.
Simplifying Expressions FAQs
What should I bear in mind when simplifying expressions?
When simplifying algebraic expressions, keep in mind that you are required to follow PEMDAS, the exponential rule, and the distributive property rules in addition to the principle of multiplication of algebraic expressions. In the end, make sure that every term on your expression is in its lowest form.
How are simplifying expressions and solving equations different?
Simplifying and solving equations are quite 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 solve them.
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