Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can appear to be scary for budding students in their first years of high school or college.
However, understanding how to process these equations is critical because it is foundational knowledge that will help them eventually be able to solve higher math and complex problems across multiple industries.
This article will share everything you must have to know simplifying expressions. We’ll learn the proponents of simplifying expressions and then verify our comprehension via some practice questions.
How Do You Simplify Expressions?
Before learning how to simplify expressions, you must understand what expressions are in the first place.
In arithmetics, expressions are descriptions that have no less than two terms. These terms can combine numbers, variables, or both and can be connected through addition or subtraction.
As an example, let’s go over the following expression.
8x + 2y - 3
This expression combines three terms; 8x, 2y, and 3. The first two terms consist of both numbers (8 and 2) and variables (x and y).
Expressions consisting of variables, coefficients, and sometimes constants, are also called polynomials.
Simplifying expressions is crucial because it opens up the possibility of grasping how to solve them. Expressions can be written in intricate ways, and without simplifying them, anyone will have a difficult time attempting to solve them, with more possibility for solving them incorrectly.
Of course, every expression differ regarding how they're simplified based on what terms they include, but there are common steps that can be applied 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, an abbreviation for 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 using addition or subtracting. If there are terms just outside the parentheses, use the distributive property to apply multiplication the term outside with the one inside.
Exponents. Where possible, use the exponent rules to simplify the terms that have exponents.
Multiplication and Division. If the equation requires it, use multiplication and division to simplify like terms that are applicable.
Addition and subtraction. Finally, add or subtract the simplified terms of the equation.
Rewrite. Ensure that there are no more like terms that require simplification, and then rewrite the simplified equation.
The Requirements For Simplifying Algebraic Expressions
Beyond the PEMDAS principle, there are a few more properties you need to 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 x as it is.
Parentheses containing 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 on the inside, as shown here: a(b+c) = ab + ac.
An extension of the distributive property is called the property of multiplication. When two stand-alone expressions within parentheses are multiplied, the distributive principle kicks in, and each individual term will have to be multiplied by the other terms, making each set of equations, common factors of one another. 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 on the inside of 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 be distributed to the terms on the inside. Despite that, this means that you should 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 prior rules were easy enough to implement as they only applied to rules that impact simple terms with numbers and variables. Still, there are additional rules that you must implement when working with expressions with exponents.
In this section, we will talk about the principles of exponents. Eight principles affect how we deal with exponentials, those 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 change in 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 by each other, their quotient applies subtraction to their respective exponents. This is written as the formula am/an = am-n.
Negative Exponents Rule. Any term with a negative exponent is equivalent 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 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 unique variables should be applied to the respective 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 denotes that any term multiplied by an expression within parentheses needs 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 like with exponents, expressions with fractions also have several rules that you have to follow.
When an expression has 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 one at a time by their numerators and denominators.
Laws of exponents. This shows us that fractions will usually be the power of the quotient rule, which will apply subtraction to the exponents of the denominators and numerators.
Simplification. Only fractions at their lowest should be expressed in the expression. Use the PEMDAS property and ensure that no two terms have the same variables.
These are the exact 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 must be noted first are the distributive property and the PEMDAS rule. The distributive property will distribute 4 to the expressions inside of the parentheses, while PEMDAS will govern the order of simplification.
Because of the distributive property, the term on the 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 each term should be in its most simplified 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 states that the you should begin with expressions inside parentheses, and in this example, that expression also requires the distributive property. Here, the term y/4 should be distributed amongst the two terms inside the parentheses, as seen in this example.
1/3x + y/4(5x) + y/4(2)
Here, let’s put aside the first term for the moment 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 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 used 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, 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 simplify, this becomes our final answer.
Simplifying Expressions FAQs
What should I bear in mind when simplifying expressions?
When simplifying algebraic expressions, keep in mind that you are required to follow the exponential rule, the distributive property, and PEMDAS rules and the concept of multiplication of algebraic expressions. Ultimately, make sure that every term on your expression is in its most simplified form.
How does solving equations differ from simplifying expressions?
Solving equations and simplifying expressions are very different, but, they can be incorporated into the same process the same process due to the fact that you first need to simplify expressions before you solve them.
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