Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions are one of the most challenging for new learners in their first years of college or even in high school.
However, grasping how to handle these equations is important because it is foundational knowledge that will help them navigate higher arithmetics and complex problems across different industries.
This article will go over everything you must have to know simplifying expressions. We’ll review the principles of simplifying expressions and then test our skills via some practice problems.
How Does Simplifying Expressions Work?
Before you can be taught how to simplify expressions, 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 contain variables, numbers, or both and can be linked through addition or subtraction.
To give an example, let’s review the following expression.
8x + 2y - 3
This expression includes three terms; 8x, 2y, and 3. The first two terms include 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 complicated ways, and without simplification, you will have a tough time attempting to solve them, with more chance for a mistake.
Obviously, each expression be different regarding how they are simplified based on what terms they incorporate, but there are common 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 refered to as the PEMDAS rule, or parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule states that the order of operations for expressions.
Parentheses. Solve equations within the parentheses first by using addition or applying 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, utilize multiplication and division to simplify like terms that apply.
Addition and subtraction. Lastly, add or subtract the remaining terms in the equation.
Rewrite. Make sure that there are no more like terms to simplify, and rewrite the simplified equation.
The Rules For Simplifying Algebraic Expressions
In addition to the PEMDAS sequence, there are a few more principles you must 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 applying addition to the coefficients 8 and 2 and leaving the x as it is.
Parentheses that contain another expression on the outside of them need to use the distributive property. The distributive property allows 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 known as the property of multiplication. When two distinct expressions within parentheses are multiplied, the distributive property kicks in, and each separate term will will require multiplication by the other terms, making 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 outside an expression in parentheses indicates that the negative expression will also need to have distribution applied, changing the signs of the terms inside the parentheses. Like in this example: -(8x + 2) will turn into -8x - 2.
Similarly, a plus sign on the outside of the parentheses denotes that it will be distributed to the terms inside. However, this means that you should eliminate 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 previous properties were simple enough to use as they only applied to principles that impact simple terms with variables and numbers. Despite that, there are additional rules that you have to follow when dealing with exponents and expressions.
Here, we will discuss the principles of exponents. 8 principles affect how we deal with exponents, that includes the following:
Zero Exponent Rule. This principle states that any term with a 0 exponent is equal to 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 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 the same variables are divided by each other, their quotient will subtract their two respective exponents. This is written 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 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 possess unique 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 assume the exponent given, (a/b)m = am/bm.
How to Simplify Expressions with the Distributive Property
The distributive property is the principle that shows us that any term multiplied by an expression within parentheses needs be multiplied by all of the expressions within. Let’s see the distributive property in action 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 contain fractions, and just as with exponents, expressions with fractions also have several rules that you must 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 separately by their numerators and denominators.
Laws of exponents. This states that fractions will usually be the power of the quotient rule, which will subtract the exponents of the denominators and numerators.
Simplification. Only fractions at their lowest should be included in the expression. Refer to the PEMDAS principle and be sure that no two terms contain the same variables.
These are the same principles that you can apply when simplifying any real numbers, whether they are decimals, square roots, binomials, linear equations, quadratic equations, and even logarithms.
Sample Questions 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 other expressions inside the parentheses, while PEMDAS will decide on the order of simplification.
Due to the distributive property, the term outside the parentheses will be multiplied by the individual 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 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 the order should start with expressions on the inside of parentheses, and in this example, that expression also requires the distributive property. In this example, the term y/4 will need to be distributed to the two terms inside the parentheses, as follows.
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. Because we know from PEMDAS that fractions require multiplication of their numerators and denominators separately, we will then have:
y/4 * 5x/1
The expression 5x/1 is used to keep things simple 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 all terms 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, which means 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 more like terms to apply simplification to, this becomes our final answer.
Simplifying Expressions FAQs
What should I remember when simplifying expressions?
When simplifying algebraic expressions, remember that you are required to obey the distributive property, PEMDAS, and the exponential rule rules as well as 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 equations and simplifying expressions are vastly different, although, they can be incorporated into the same process the same process due to the fact that you first need to simplify expressions before you begin solving them.
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