Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can appear to be scary for beginner learners in their primary years of college or even in high school.
Still, understanding how to process these equations is critical because it is primary information that will help them eventually be able to solve higher arithmetics and complex problems across various industries.
This article will go over everything you should review to learn simplifying expressions. We’ll cover the laws of simplifying expressions and then test our comprehension with some practice questions.
How Do I Simplify an Expression?
Before you can learn how to simplify them, you must grasp what expressions are at their core.
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 subtraction or addition.
As an example, let’s take a look at the following expression.
8x + 2y - 3
This expression combines three terms; 8x, 2y, and 3. The first two consist of both numbers (8 and 2) and variables (x and y).
Expressions containing variables, coefficients, and sometimes constants, are also referred to as polynomials.
Simplifying expressions is crucial because it lays the groundwork for understanding how to solve them. Expressions can be written in complicated ways, and without simplifying them, you will have a tough time attempting to solve them, with more possibility for error.
Obviously, every expression be different regarding how they are simplified based on what terms they include, but there are typical steps that can be applied to all rational expressions of real numbers, whether they are logarithms, square roots, etc.
These steps are known as the PEMDAS rule, or parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule declares the order of operations for expressions.
Parentheses. Solve equations within the parentheses first by applying addition or applying subtraction. 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 principles to simplify the terms that have exponents.
Multiplication and Division. If the equation requires it, utilize multiplication or division rules to simplify like terms that are applicable.
Addition and subtraction. Lastly, use addition or subtraction the resulting terms of the equation.
Rewrite. Ensure that there are no additional like terms to simplify, and then rewrite the simplified equation.
The Requirements For Simplifying Algebraic Expressions
Beyond the PEMDAS rule, there are a few additional rules you must be informed 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 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 on the inside, or as follows: a(b+c) = ab + ac.
An extension of the distributive property is referred to as the property of multiplication. When two distinct expressions within parentheses are multiplied, the distribution principle kicks in, and all unique term will have 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 directly outside of an expression in parentheses indicates that the negative expression should also need to have distribution applied, changing the signs of the terms inside the parentheses. For example: -(8x + 2) will turn into -8x - 2.
Similarly, a plus sign outside the parentheses means 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 owing 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 follow as they only applied to properties that affect simple terms with numbers and variables. Despite that, there are a few other rules that you have to implement when working with exponents and expressions.
Here, we will talk about the principles of exponents. 8 properties impact how we utilize exponentials, which are the following:
Zero Exponent Rule. This property states that any term with a 0 exponent is equal to 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 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 applies subtraction to their two 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 already with 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 required 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.
Simplifying Expressions with the Distributive Property
The distributive property is the principle that shows us that any term multiplied by an expression within parentheses must be multiplied by all of the expressions within. Let’s watch 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.
How to Simplify Expressions with Fractions
Certain expressions can consist of fractions, and just as with exponents, expressions with fractions also have multiple rules that you have to follow.
When an expression contains fractions, here is what to remember.
Distributive property. The distributive property a(b+c) = ab + ac, when applied to fractions, will multiply fractions one at a time by their denominators and numerators.
Laws of exponents. This states that fractions will more likely 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 expressed in the expression. Refer to the PEMDAS principle and make sure that no two terms contain the same variables.
These are the same properties that you can apply when simplifying any real numbers, whether they are square roots, binomials, decimals, logarithms, linear equations, or quadratic equations.
Sample Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this case, the principles that need to be noted first are the PEMDAS and the distributive property. The distributive property will distribute 4 to all other expressions inside the parentheses, while PEMDAS will decide on the order of simplification.
Due to 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, remember to add the terms with matching 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 first in order should be expressions inside parentheses, and in this example, that expression also necessitates the distributive property. Here, the term y/4 should be distributed to the two terms within the parentheses, as seen here.
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. Since we know from PEMDAS that fractions will require multiplication of their numerators and denominators separately, we will then have:
y/4 * 5x/1
The expression 5x/1 is used for simplicity 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 all terms 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 like 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 apply simplification to, this becomes our final answer.
Simplifying Expressions FAQs
What should I remember when simplifying expressions?
When simplifying algebraic expressions, keep in mind that you have to follow the distributive property, PEMDAS, and the exponential rule rules as well as the concept of multiplication of algebraic expressions. Finally, ensure 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, but, they can be incorporated into the same process the same process due to the fact that you have to simplify expressions before you begin solving them.
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