Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be challenging for beginner learners in their first years of high school or college.
Still, grasping how to handle these equations is essential because it is basic information that will help them eventually be able to solve higher math and complex problems across different industries.
This article will discuss everything you need to learn simplifying expressions. We’ll learn the proponents of simplifying expressions and then test our skills via some practice questions.
How Do I Simplify an Expression?
Before you can be taught how to simplify them, you must learn what expressions are at their core.
In arithmetics, expressions are descriptions that have a minimum of two terms. These terms can include variables, numbers, or both and can be connected through addition or subtraction.
As an example, let’s take a look at 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 variables, coefficients, and sometimes constants, are also called polynomials.
Simplifying expressions is crucial because it paves the way for understanding how to solve them. Expressions can be expressed in convoluted ways, and without simplification, everyone will have a hard time attempting to solve them, with more possibility for solving them incorrectly.
Of course, every expression vary concerning how they're simplified based on what terms they contain, but there are common 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, or parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule states that the order of operations for expressions.
Parentheses. Resolve equations between the parentheses first by using addition or applying subtraction. If there are terms just outside the parentheses, use the distributive property to multiply the term on the outside with the one inside.
Exponents. Where possible, use the exponent properties to simplify the terms that contain exponents.
Multiplication and Division. If the equation requires it, utilize multiplication or division rules to simplify like terms that are applicable.
Addition and subtraction. Then, add or subtract the remaining terms of the equation.
Rewrite. Ensure that there are no additional like terms that require simplification, and then rewrite the simplified equation.
The Requirements For Simplifying Algebraic Expressions
Along with the PEMDAS rule, there are a few additional properties you need to be informed of when working 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 retaining the variable x as it is.
Parentheses containing 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 on the inside, or as follows: a(b+c) = ab + ac.
An extension of the distributive property is known as the property of multiplication. When two separate expressions within parentheses are multiplied, the distribution rule applies, and every separate term will need 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 have distribution applied, changing the signs of the terms on the inside of the parentheses. As is the case in this example: -(8x + 2) will turn into -8x - 2.
Similarly, a plus sign right outside the parentheses will mean that it will have distribution applied to the terms on the 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 rules were easy enough to implement as they only dealt with properties that impact simple terms with numbers and variables. Still, there are more rules that you have to apply when dealing with exponents and expressions.
In this section, we will talk about the properties of exponents. 8 principles influence how we deal with exponents, that includes the following:
Zero Exponent Rule. This principle states that any term with the exponent of 0 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 matching variables are divided by each other, their quotient subtracts their two respective 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 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 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 take the exponent given, (a/b)m = am/bm.
How to Simplify Expressions with the Distributive Property
The distributive property is the property that says that any term multiplied by an expression on the inside of a parentheses must be multiplied by all of the expressions inside. Let’s witness 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.
Simplifying Expressions with Fractions
Certain expressions can consist of fractions, and just like with exponents, expressions with fractions also have some rules that you must follow.
When an expression has fractions, here is what to keep in mind.
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 shows us that fractions will typically be the power of the quotient rule, which will subtract the exponents of the numerators and denominators.
Simplification. Only fractions at their lowest form should be written in the expression. Use the PEMDAS principle and ensure that no two terms share 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 Examples for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
Here, the rules that need to be noted first are the PEMDAS and the distributive property. The distributive property will distribute 4 to all the expressions on the inside of the parentheses, while PEMDAS will govern the order of simplification.
Due to the distributive property, the term outside of 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 each 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 states that the first in order should be expressions on the inside of parentheses, and in this case, that expression also requires the distributive property. In this example, the term y/4 must be distributed to the two terms within the parentheses, as seen in this example.
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 associated with them. Remember 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 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 utilized to distribute all terms 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 like 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 keep in mind when simplifying expressions?
When simplifying algebraic expressions, keep in mind that you have to obey PEMDAS, the exponential rule, and the distributive property rules and the rule of multiplication of algebraic expressions. Ultimately, ensure that every term on your expression is in its lowest form.
What is the difference between solving an equation and simplifying an expression?
Solving and simplifying expressions are very different, however, they can be combined the same process because you must first simplify expressions before solving them.
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