Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions are one of the most scary for new learners in their primary years of college or even in high school.
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 math and complex problems across multiple industries.
This article will discuss everything you need to learn simplifying expressions. We’ll review the laws of simplifying expressions and then verify what we've learned through some sample questions.
How Does Simplifying Expressions Work?
Before you can learn how to simplify expressions, you must understand what expressions are in the first place.
In mathematics, expressions are descriptions that have at least two terms. These terms can combine numbers, variables, or both and can be linked through addition or subtraction.
For example, let’s go over 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 coefficients, variables, and occasionally constants, are also known as polynomials.
Simplifying expressions is crucial because it lays the groundwork for understanding how to solve them. Expressions can be written in intricate ways, and without simplifying them, anyone will have a difficult time trying to solve them, with more opportunity for error.
Obviously, every expression vary regarding how they are simplified based on what terms they contain, but there are common steps that apply to all rational expressions of real numbers, whether they are square roots, logarithms, or otherwise.
These steps are called the PEMDAS rule, short for parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule declares the order of operations for expressions.
Parentheses. Simplify equations between the parentheses first by using addition or subtracting. If there are terms right outside the parentheses, use the distributive property to multiply the term outside with the one on the inside.
Exponents. Where workable, use the exponent properties to simplify the terms that contain exponents.
Multiplication and Division. If the equation necessitates it, utilize multiplication or division rules to simplify like terms that apply.
Addition and subtraction. Then, add or subtract the simplified terms in the equation.
Rewrite. Ensure that there are no remaining like terms to simplify, and rewrite the simplified equation.
Here are the Properties For Simplifying Algebraic Expressions
In addition to the PEMDAS principle, there are a few more properties you need to be informed of when simplifying 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 applying addition to the coefficients 8 and 2 and retaining the x as it is.
Parentheses that include another expression directly outside of them need to apply the distributive property. The distributive property gives you the ability to 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 distinct expressions within parentheses are multiplied, the distributive rule is applied, and every individual term will need to be multiplied by the other terms, resulting in each set of equations, common factors of one another. Like in this example: (a + b)(c + d) = a(c + d) + b(c + d).
A negative sign directly outside of an expression in parentheses denotes that the negative expression should also need to have distribution applied, changing the signs of the terms inside the parentheses. As is the case in this example: -(8x + 2) will turn into -8x - 2.
Likewise, a plus sign right outside the parentheses denotes that it will have distribution applied to the terms on the inside. Despite that, this means that you are able to eliminate the parentheses and write the expression as is due to the fact that the plus sign doesn’t change anything when distributed.
How to Simplify Expressions with Exponents
The previous rules were simple enough to use as they only applied to properties that affect simple terms with numbers and variables. Still, there are a few other rules that you need to implement when working with exponents and expressions.
Next, we will review the principles of exponents. Eight properties influence how we deal with exponentials, which are the following:
Zero Exponent Rule. This property states that any term with a 0 exponent equals 1. Or a0 = 1.
Identity Exponent Rule. Any term with the exponent of 1 won't alter the value. Or a1 = a.
Product Rule. When two terms with the same variables are multiplied, 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 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 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 applied to it, or (am)n = amn.
Power of a Product Rule. An exponent applied to two terms that have 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 assume 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 on the inside of a parentheses must be multiplied by all of the expressions within. 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 result is 6x + 10.
How to Simplify Expressions with Fractions
Certain expressions contain fractions, and just as with exponents, expressions with fractions also have multiple rules that you must 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 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 state should be written in the expression. Use the PEMDAS property and be sure that no two terms share matching variables.
These are the same rules 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 properties that should be noted first are the PEMDAS and the distributive property. The distributive property will distribute 4 to all other expressions inside of the parentheses, while PEMDAS will dictate the order of simplification.
Due to the distributive property, the term outside 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, remember to add all the terms with the same variables, and all term should be in its lowest form.
28x + 28 - 3y
Rearrange the equation as follows:
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 case, that expression also needs the distributive property. In this example, the term y/4 will need to be distributed within 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 associated with 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 used to distribute every term 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, meaning 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 keep in mind when simplifying expressions?
When simplifying algebraic expressions, bear in mind that you must obey PEMDAS, the exponential rule, and the distributive property rules and the rule of multiplication of algebraic expressions. In the end, ensure that every term on your expression is in its most simplified form.
How does solving equations differ from simplifying expressions?
Simplifying and solving equations are vastly different, however, they can be part of the same process the same process due to the fact that you must first simplify expressions before solving them.
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