Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can appear to be challenging for budding learners in their early years of college or even in high school.
Still, grasping how to handle these equations is critical because it is basic information that will help them eventually be able to solve higher math and advanced problems across multiple industries.
This article will discuss everything you should review to learn simplifying expressions. We’ll review the laws of simplifying expressions and then verify our comprehension with some sample problems.
How Do I Simplify an Expression?
Before you can learn how to simplify expressions, you must learn what expressions are to begin with.
In mathematics, expressions are descriptions that have no less than two terms. These terms can include variables, numbers, or both and can be linked 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 terms include 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 important because it lays the groundwork for grasping how to solve them. Expressions can be written in convoluted ways, and without simplifying them, you will have a difficult time trying to solve them, with more opportunity for a mistake.
Undoubtedly, every expression differ concerning how they are 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 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. Simplify equations inside the parentheses first by adding or subtracting. If there are terms just outside the parentheses, use the distributive property to multiply the term outside with the one inside.
Exponents. Where workable, use the exponent properties to simplify the terms that have exponents.
Multiplication and Division. If the equation calls for it, use the multiplication and division principles to simplify like terms that are applicable.
Addition and subtraction. Lastly, use addition or subtraction the remaining terms in the equation.
Rewrite. Ensure that there are no additional like terms to simplify, and then rewrite the simplified equation.
Here are the Rules For Simplifying Algebraic Expressions
Beyond the PEMDAS sequence, there are a few additional rules you need to be aware of when simplifying algebraic expressions.
You can only apply simplification to 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 x as it is.
Parentheses that contain another expression on the outside of them need to utilize the distributive property. The distributive property prompts you to simplify terms 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 principle of multiplication. When two distinct expressions within parentheses are multiplied, the distributive principle is applied, and every separate term will will require multiplication by the other terms, resulting in each set of equations, common factors of each other. For example: (a + b)(c + d) = a(c + d) + b(c + d).
A negative sign directly outside of an expression in parentheses indicates that the negative expression will also need to have distribution applied, changing the signs of the terms inside the parentheses. Like in this example: -(8x + 2) will turn into -8x - 2.
Similarly, a plus sign on the outside of the parentheses means that it will have distribution applied to the terms on the inside. However, this means that you are able to remove the parentheses and write the expression as is because the plus sign doesn’t alter anything when distributed.
How to Simplify Expressions with Exponents
The previous properties were simple enough to implement as they only dealt with rules that impact simple terms with variables and numbers. Still, there are additional rules that you need to apply when working with expressions with exponents.
Here, we will talk about the properties of exponents. 8 principles influence how we deal with exponentials, which 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 the exponent of 1 won't alter the value. Or a1 = a.
Product Rule. When two terms with equivalent 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 matching variables are divided by each other, their quotient applies subtraction to 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 already with 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 possess different variables will 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 take the exponent given, (a/b)m = am/bm.
How to Simplify Expressions with the Distributive Property
The distributive property is the principle that denotes that any term multiplied by an expression within parentheses should 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 result is 6x + 10.
How to Simplify Expressions with Fractions
Certain expressions can consist of fractions, and just as with exponents, expressions with fractions also have several rules that you have to follow.
When an expression consist of 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 shows 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 should be included in the expression. Use the PEMDAS principle and ensure that no two terms contain matching variables.
These are the exact rules that you can apply when simplifying any real numbers, whether they are binomials, decimals, square roots, logarithms, linear equations, or quadratic equations.
Sample Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this example, 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, 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 like this:
28x - 3y + 28
Example 2
Simplify the expression 1/3x + y/4(5x + 2)
The PEMDAS rule expresses that the the order should start with expressions within parentheses, and in this scenario, that expression also needs the distributive property. In this example, the term y/4 will need to be distributed amongst the two terms inside 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 associated with them. Since we know from PEMDAS that fractions require multiplication of their denominators and numerators separately, we will then have:
y/4 * 5x/1
The expression 5x/1 is used to keep things simple 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 utilized to distribute each 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, which tells us 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 more like terms to be simplified, 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 the distributive property, PEMDAS, and the exponential rule rules and the rule of multiplication of algebraic expressions. Finally, make sure that every term on your expression is in its lowest form.
How are simplifying expressions and solving equations different?
Solving equations and simplifying expressions are vastly different, however, they can be incorporated into the same process the same process because you first need to simplify expressions before you solve them.
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