Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions are one of the most challenging for beginner learners in their primary years of high school or college.
Still, understanding how to handle these equations is essential because it is foundational information that will help them move on to higher mathematics and advanced problems across various industries.
This article will go over everything you need to master simplifying expressions. We’ll learn the proponents of simplifying expressions and then validate our comprehension through some sample questions.
How Do You Simplify Expressions?
Before you can be taught how to simplify expressions, you must learn what expressions are in the first place.
In mathematics, expressions are descriptions that have a minimum of two terms. These terms can include 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 consisting of variables, coefficients, and sometimes constants, are also referred to as polynomials.
Simplifying expressions is crucial because it paves the way for learning how to solve them. Expressions can be expressed in intricate ways, and without simplification, anyone will have a difficult time attempting to solve them, with more opportunity for a mistake.
Obviously, all expressions will be different regarding how they're simplified based on what terms they include, but there are common steps that are applicable 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, short for 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 applying addition or using 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 feasible, use the exponent principles to simplify the terms that include exponents.
Multiplication and Division. If the equation necessitates it, use multiplication and division to simplify like terms that are applicable.
Addition and subtraction. Lastly, use addition or subtraction the simplified terms in the equation.
Rewrite. Ensure that there are no additional like terms to simplify, and then rewrite the simplified equation.
Here are the Properties For Simplifying Algebraic Expressions
In addition to the PEMDAS sequence, there are a few more rules you must be informed 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 keep 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 leaving 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 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 separate expressions within parentheses are multiplied, the distributive property kicks in, and each unique term will will require multiplication by the other terms, resulting in each set of equations, common factors of one another. For example: (a + b)(c + d) = a(c + d) + b(c + d).
A negative sign right outside of an expression in parentheses denotes that the negative expression must also need to be distributed, changing the signs of the terms on the inside of the parentheses. For example: -(8x + 2) will turn into -8x - 2.
Likewise, a plus sign right outside the parentheses denotes that it will be distributed to the terms on the inside. However, this means that you can remove 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 straight-forward enough to follow as they only dealt with properties that impact simple terms with numbers and variables. However, there are additional rules that you need to implement when dealing with expressions with exponents.
In this section, we will review the principles of exponents. 8 rules influence how we deal with exponents, those are the following:
Zero Exponent Rule. This rule states that any term with a 0 exponent is equivalent to 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 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 applies subtraction to their applicable exponents. This is seen 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 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 differing variables should be applied to the respective 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 rule that shows us that any term multiplied by an expression on the inside of a parentheses needs be multiplied by all of the expressions on the inside. Let’s see the distributive property used 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 as with exponents, expressions with fractions also have some rules that you need to follow.
When an expression includes fractions, here's 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 tells us 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 state should be written in the expression. Refer to the PEMDAS property and be sure that no two terms contain matching variables.
These are the same properties that you can apply when simplifying any real numbers, whether they are binomials, decimals, square roots, linear equations, quadratic equations, and even logarithms.
Sample Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
Here, the properties that must be noted first are the distributive property and the PEMDAS rule. The distributive property will distribute 4 to all the expressions on the inside of the parentheses, while PEMDAS will dictate the order of simplification.
As a result of the distributive property, the term outside of the parentheses will be multiplied by each term on the inside.
4(2x) + 4(5x) + 4(7) - 3y
8x + 20x + 28 - 3y
When simplifying equations, you should add all the terms with the same variables, and all term should be in its lowest form.
28x + 28 - 3y
Rearrange the equation this way:
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 on the inside of parentheses, and in this scenario, that expression also requires the distributive property. In this scenario, the term y/4 should be distributed within the two terms within the parentheses, as seen in this example.
1/3x + y/4(5x) + y/4(2)
Here, let’s set aside the first term for the moment and simplify the terms with factors assigned to them. Remember we know from PEMDAS that fractions will require multiplication of their denominators and numerators individually, we will then have:
y/4 * 5x/1
The expression 5x/1 is used to keep things simple since 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 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
Due to the fact that there are no remaining like terms to be simplified, this becomes our final answer.
Simplifying Expressions FAQs
What should I bear in mind when simplifying expressions?
When simplifying algebraic expressions, remember that you have to obey the exponential rule, the distributive property, and PEMDAS rules and the principle of multiplication of algebraic expressions. In the end, 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 equations and simplifying expressions are vastly different, although, they can be part of the same process the same process because you have to simplify expressions before you solve them.
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