Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be intimidating for budding students in their first years of high school or college.
However, grasping how to deal with these equations is essential because it is foundational information that will help them eventually be able to solve higher arithmetics and advanced problems across multiple industries.
This article will discuss everything you must have to master simplifying expressions. We’ll cover the laws of simplifying expressions and then validate what we've learned with some sample problems.
How Do I Simplify an Expression?
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 numbers, variables, or both and can be connected through addition or subtraction.
To give an example, let’s go over the following expression.
8x + 2y - 3
This expression contains three terms; 8x, 2y, and 3. The first two include 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 crucial because it opens up the possibility of understanding how to solve them. Expressions can be written in complicated ways, and without simplification, everyone will have a hard time attempting to solve them, with more opportunity for a mistake.
Of course, each expression vary in how they're simplified based on what terms they contain, but there are general steps that apply to all rational expressions of real numbers, regardless of whether they are logarithms, square roots, etc.
These steps are refered to as the PEMDAS rule, short for parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule shows us the order of operations for expressions.
Parentheses. Resolve equations within the parentheses first by applying addition or subtracting. 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 feasible, use the exponent rules to simplify the terms that contain exponents.
Multiplication and Division. If the equation calls for it, use multiplication or division rules to simplify like terms that apply.
Addition and subtraction. Lastly, use addition or subtraction the simplified terms in the equation.
Rewrite. Ensure that there are no remaining like terms that require simplification, and rewrite the simplified equation.
Here are the Requirements For Simplifying Algebraic Expressions
Along with the PEMDAS principle, there are a few more rules you should be informed of when dealing with algebraic expressions.
You can only simplify terms with common variables. When applying addition to these terms, add the coefficient numbers and maintain the variables as [[is|they are]-70. For example, the expression 8x + 2x can be simplified to 10x by adding coefficients 8 and 2 and keeping the variable x as it is.
Parentheses that contain another expression on the outside of them need to apply the distributive property. The distributive property allows you to simplify terms on the outside of parentheses by distributing them to the terms inside, or as follows: 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 distribution rule kicks in, and all unique term will need to be multiplied by the other terms, resulting in each set of equations, common factors of each other. Such as is the case here: (a + b)(c + d) = a(c + d) + b(c + d).
A negative sign right outside of 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. Like in this example: -(8x + 2) will turn into -8x - 2.
Likewise, a plus sign right outside the parentheses will mean that it will be distributed to the terms on the inside. But, this means that you are able to remove the parentheses and write the expression as is owing to the fact that the plus sign doesn’t alter anything when distributed.
How to Simplify Expressions with Exponents
The prior properties were simple enough to implement as they only applied to rules that affect simple terms with numbers and variables. However, there are additional rules that you have to follow when working with exponents and expressions.
Next, we will talk about the principles of exponents. 8 principles affect how we process exponentials, that includes 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 the exponent of 1 will not alter the value. Or a1 = a.
Product Rule. When two terms with the same 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 will subtract their respective exponents. This is written as 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 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 that already has an exponent, the term will result in 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 numerator and denominator will assume the exponent given, (a/b)m = am/bm.
Simplifying Expressions with the Distributive Property
The distributive property is the property that denotes that any term multiplied by an expression within parentheses must be multiplied by all of the expressions within. 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 expression then becomes 6x + 10.
How to Simplify Expressions with Fractions
Certain expressions can consist of fractions, and just as with exponents, expressions with fractions also have some rules that you have to follow.
When an expression contains 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 states that fractions will more likely be the power of the quotient rule, which will apply subtraction to the exponents of the numerators and denominators.
Simplification. Only fractions at their lowest should be expressed in the expression. Refer to the PEMDAS property and be sure that no two terms share matching variables.
These are the exact principles that you can apply when simplifying any real numbers, whether they are binomials, decimals, square roots, linear equations, quadratic equations, and even logarithms.
Practice Examples for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this example, the properties that should be noted first are PEMDAS and the distributive property. The distributive property will distribute 4 to the expressions inside the parentheses, while PEMDAS will govern the order of simplification.
Because of 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 the terms with the same variables, and all 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 expresses that the you should begin with expressions within parentheses, and in this case, that expression also necessitates the distributive property. In this scenario, the term y/4 will need to be distributed within 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 attached 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 for simplicity 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 used 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 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
Because there are no other like terms to be simplified, this becomes our final answer.
Simplifying Expressions FAQs
What should I remember when simplifying expressions?
When simplifying algebraic expressions, remember that you have to follow the exponential rule, the distributive property, and PEMDAS rules and the concept 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 very different, although, they can be incorporated into the same process the same process since you must first simplify expressions before you solve them.
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