Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be challenging for new learners in their primary years of high school or college.
However, understanding how to handle these equations is critical because it is primary knowledge that will help them navigate higher math and advanced problems across different industries.
This article will discuss everything you must have to master simplifying expressions. We’ll learn the principles of simplifying expressions and then validate our comprehension via some sample questions.
How Do You Simplify Expressions?
Before you can learn how to simplify them, you must learn what expressions are at their core.
In mathematics, expressions are descriptions that have no less than two terms. These terms can contain variables, numbers, or both and can be linked through addition or subtraction.
As an example, let’s review the following expression.
8x + 2y - 3
This expression includes three terms; 8x, 2y, and 3. The first two terms contain both numbers (8 and 2) and variables (x and y).
Expressions that include coefficients, variables, and occasionally constants, are also referred to as polynomials.
Simplifying expressions is essential because it lays the groundwork for grasping how to solve them. Expressions can be written in intricate ways, and without simplifying them, anyone will have a tough time trying to solve them, with more possibility for error.
Of course, all expressions will differ concerning how they're simplified depending on what terms they contain, 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 known as the PEMDAS rule, short for parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule states that the order of operations for expressions.
Parentheses. Simplify equations within the parentheses first by applying addition or applying subtraction. If there are terms just outside the parentheses, use the distributive property to apply multiplication the term outside with the one on the inside.
Exponents. Where possible, use the exponent rules to simplify the terms that contain exponents.
Multiplication and Division. If the equation necessitates it, use multiplication or division rules to simplify like terms that apply.
Addition and subtraction. Lastly, use addition or subtraction the resulting terms of the equation.
Rewrite. Ensure that there are no more like terms that require simplification, and rewrite the simplified equation.
The Requirements For Simplifying Algebraic Expressions
Beyond the PEMDAS sequence, there are a few additional rules you need to 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 retaining the variable x as it is.
Parentheses that include another expression on the 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 inside, for example: a(b+c) = ab + ac.
An extension of the distributive property is referred to as the principle of multiplication. When two separate expressions within parentheses are multiplied, the distributive rule kicks in, and each separate term will will require multiplication by the other terms, making each set of equations, common factors of each other. For example: (a + b)(c + d) = a(c + d) + b(c + d).
A negative sign outside an expression in parentheses means that the negative expression will also need to have distribution applied, changing the signs of the terms on the inside of the parentheses. Like in this example: -(8x + 2) will turn into -8x - 2.
Likewise, a plus sign right outside the parentheses will mean that it will have distribution applied to the terms on the inside. Despite that, 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 prior properties were straight-forward enough to use as they only dealt with properties that affect simple terms with numbers and variables. However, there are additional rules that you have to implement when working with exponents and expressions.
Next, we will review the principles of exponents. 8 principles impact how we utilize exponentials, those are the following:
Zero Exponent Rule. This principle states that any term with the exponent of 0 equals 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 multiplied by each other, 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, their quotient subtracts their 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 that already has 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 needs to 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 rule that states that any term multiplied by an expression within parentheses needs be multiplied by all of the expressions inside. Let’s watch 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 like with exponents, expressions with fractions also have some rules that you must follow.
When an expression has 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 numerators and denominators.
Laws of exponents. This tells us that fractions will typically 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 written in the expression. Refer to the PEMDAS principle and be sure that no two terms possess matching variables.
These are the exact properties that you can apply when simplifying any real numbers, whether they are square roots, binomials, decimals, quadratic equations, logarithms, or linear equations.
Practice Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this case, the properties that need to be noted first are the distributive property and the PEMDAS rule. The distributive property will distribute 4 to all other expressions on the inside of the parentheses, while PEMDAS will govern the order of simplification.
Because of the distributive property, the term on the 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, be sure to add all the terms with matching variables, and each term should be in its lowest 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 you should begin with expressions within parentheses, and in this case, that expression also requires the distributive property. In this scenario, the term y/4 will need to be distributed to the two terms inside 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. 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 used to distribute each 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 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 remember when simplifying expressions?
When simplifying algebraic expressions, remember that you must obey the distributive property, PEMDAS, and the exponential rule rules as well as the principle of multiplication of algebraic expressions. Finally, make sure that every term on your expression is in its lowest form.
What is the difference between solving an equation and simplifying an expression?
Simplifying and solving equations are vastly different, although, 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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