Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can appear to be challenging for beginner pupils in their primary years of high school or college.
Nevertheless, learning how to deal with these equations is important because it is primary knowledge that will help them navigate higher mathematics and advanced problems across different industries.
This article will share everything you need to know simplifying expressions. We’ll review the proponents of simplifying expressions and then verify what we've learned with some sample problems.
How Does Simplifying Expressions Work?
Before learning how to simplify expressions, you must understand what expressions are in the first place.
In mathematics, 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 review the following expression.
8x + 2y - 3
This expression combines three terms; 8x, 2y, and 3. The first two contain both numbers (8 and 2) and variables (x and y).
Expressions consisting of coefficients, variables, and occasionally constants, are also known as polynomials.
Simplifying expressions is essential because it paves the way for understanding how to solve them. Expressions can be written in convoluted ways, and without simplifying them, anyone will have a tough time attempting to solve them, with more opportunity for solving them incorrectly.
Undoubtedly, every expression differ regarding how they're simplified depending on what terms they incorporate, but there are common steps that apply to all rational expressions of real numbers, regardless of whether they are square roots, logarithms, or otherwise.
These steps are called the PEMDAS rule, short for parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule declares the order of operations for expressions.
Parentheses. Simplify equations between the parentheses first by adding or using subtraction. If there are terms just outside the parentheses, use the distributive property to apply multiplication the term on the outside with the one inside.
Exponents. Where workable, use the exponent properties to simplify the terms that contain exponents.
Multiplication and Division. If the equation necessitates it, use the multiplication and division principles to simplify like terms that are applicable.
Addition and subtraction. Lastly, use addition or subtraction the simplified terms of the equation.
Rewrite. Ensure that there are no additional like terms to simplify, then rewrite the simplified equation.
Here are the Requirements For Simplifying Algebraic Expressions
Beyond the PEMDAS principle, there are a few more principles you should be informed of when working 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 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 inside, as shown here: a(b+c) = ab + ac.
An extension of the distributive property is called the principle of multiplication. When two stand-alone expressions within parentheses are multiplied, the distributive principle is applied, and every individual 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 indicates that the negative expression will 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 right outside the parentheses means that it will be distributed to the terms inside. However, this means that you should eliminate the parentheses and write the expression as is because the plus sign doesn’t change anything when distributed.
How to Simplify Expressions with Exponents
The prior rules were easy enough to implement as they only applied to rules that impact simple terms with numbers and variables. Despite that, there are a few other rules that you need to apply when dealing with exponents and expressions.
Here, we will discuss the properties of exponents. Eight rules impact how we process exponents, that includes the following:
Zero Exponent Rule. This rule states that any term with a 0 exponent equals 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 apply multiplication, their product will add their two exponents. This is expressed in the formula am × an = am+n
Quotient Rule. When two terms with the same variables are divided by each other, their quotient applies subtraction to their two respective exponents. This is expressed in 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 expressed with 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 differing variables will be applied to the appropriate 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 principle that denotes that any term multiplied by an expression on the inside of a parentheses must be multiplied by all of the expressions inside. 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 can consist of fractions, and just like with exponents, expressions with fractions also have some rules that you have to follow.
When an expression consist of 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 typically 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 written in the expression. Refer to 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, logarithms, linear equations, or quadratic equations.
Practice Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this case, the principles that need to be noted first are the distributive property and the PEMDAS rule. The distributive property will distribute 4 to all other expressions inside the parentheses, while PEMDAS will govern the order of simplification.
As a result of the distributive property, the term on the 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, be sure to add all the terms with matching variables, and all term should be in its most simplified 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 first in order should be expressions within parentheses, and in this case, that expression also requires the distributive property. In this example, the term y/4 should be distributed amongst the two terms on the inside of 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. Because we know from PEMDAS that fractions will need to multiply their denominators and numerators separately, 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 each 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 similar variables multiplied together and multiply their coefficients. This gives us:
24x3 + 4x2 + 18xy + 3y
Due to the fact that there are no other like terms to simplify, 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 the distributive property, PEMDAS, and the exponential rule rules in addition to the concept of multiplication of algebraic expressions. Ultimately, make sure 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, however, they can be incorporated into the same process the same process because you must first simplify expressions before you solve them.
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