Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be scary for beginner pupils in their primary years of college or even in high school.
Nevertheless, grasping how to handle these equations is important because it is basic information that will help them eventually be able to solve higher arithmetics and complex problems across different industries.
This article will share everything you must have to know simplifying expressions. We’ll review the principles of simplifying expressions and then test what we've learned with some practice questions.
How Does Simplifying Expressions Work?
Before you can be taught how to simplify them, you must grasp what expressions are to begin with.
In arithmetics, expressions are descriptions that have at least two terms. These terms can include variables, numbers, or both and can be linked through subtraction or addition.
For example, let’s review the following expression.
8x + 2y - 3
This expression combines three terms; 8x, 2y, and 3. The first two terms contain both numbers (8 and 2) and variables (x and y).
Expressions that incorporate variables, coefficients, and sometimes constants, are also referred to as polynomials.
Simplifying expressions is important because it opens up the possibility of learning how to solve them. Expressions can be expressed in convoluted ways, and without simplifying them, anyone will have a hard time trying to solve them, with more opportunity for error.
Obviously, each expression vary in how they are simplified based on what terms they contain, but there are general 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. Solve equations inside the parentheses first by using addition or applying subtraction. If there are terms right outside the parentheses, use the distributive property to multiply the term on the outside with the one inside.
Exponents. Where possible, use the exponent rules to simplify the terms that include exponents.
Multiplication and Division. If the equation requires it, use multiplication and division to simplify like terms that apply.
Addition and subtraction. Lastly, add or subtract the remaining terms in the equation.
Rewrite. Ensure that there are no remaining like terms to simplify, then rewrite the simplified equation.
The Properties For Simplifying Algebraic Expressions
In addition to the PEMDAS rule, there are a few more properties 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 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 leaving the x as it is.
Parentheses that include another expression outside of them need to use the distributive property. The distributive property allows you to simplify terms 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 referred to as the concept of multiplication. When two separate expressions within parentheses are multiplied, the distribution principle applies, and each unique term will have to be multiplied 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 indicates that the negative expression should 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 denotes that it will have distribution applied to the terms inside. However, this means that you should 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 prior rules were straight-forward enough to use as they only dealt with rules that impact simple terms with numbers and variables. However, there are additional rules that you need to apply when working with expressions with exponents.
In this section, we will discuss the principles of exponents. 8 principles impact how we utilize exponents, those are the following:
Zero Exponent Rule. This principle 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 doesn't change in value. Or a1 = a.
Product Rule. When two terms with the same variables are apply multiplication, their product will add their exponents. This is written as am × an = am+n
Quotient Rule. When two terms with matching variables are divided, their quotient will subtract 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 end up being the 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 denominator and numerator will acquire the exponent given, (a/b)m = am/bm.
Simplifying Expressions with the Distributive Property
The distributive property is the rule that says that any term multiplied by an expression within parentheses should 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 result is 6x + 10.
Simplifying Expressions with Fractions
Certain expressions contain fractions, and just like with exponents, expressions with fractions also have multiple rules that you need to follow.
When an expression includes fractions, here's 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 numerators and denominators.
Laws of exponents. This tells us that fractions will typically be the power of the quotient rule, which will subtract the exponents of the denominators and numerators.
Simplification. Only fractions at their lowest form should be expressed in the expression. Use the PEMDAS principle and be sure that no two terms share the same variables.
These are the same properties that you can apply when simplifying any real numbers, whether they are decimals, square roots, binomials, linear equations, quadratic equations, and even logarithms.
Practice Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this example, the properties that must be noted first are the distributive property and the PEMDAS rule. The distributive property will distribute 4 to the expressions inside the parentheses, while PEMDAS will govern the order of simplification.
As a result of the distributive property, the term outside of the parentheses will be multiplied by the terms inside.
4(2x) + 4(5x) + 4(7) - 3y
8x + 20x + 28 - 3y
When simplifying equations, you should add all the terms with matching variables, and every 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 states that the first in order should be expressions on the inside of parentheses, and in this example, that expression also requires the distributive property. In this scenario, the term y/4 must be distributed within 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 now and simplify the terms with factors associated with them. Since we know from PEMDAS that fractions will need to multiply their numerators and denominators individually, 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 used 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 be simplified, 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 follow the distributive property, PEMDAS, and the exponential rule 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 are simplifying expressions and solving equations different?
Solving and simplifying expressions are very different, but, they can be part of the same process the same process because you first need to simplify expressions before you solve them.
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