Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions are one of the most scary for budding students in their early years of high school or college.
However, learning how to process these equations is important because it is foundational information that will help them navigate higher arithmetics and complex problems across various industries.
This article will go over everything you should review to master simplifying expressions. We’ll cover the laws of simplifying expressions and then verify our skills through some practice problems.
How Do I Simplify an Expression?
Before you can learn how to simplify expressions, you must grasp what expressions are to begin with.
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 subtraction or addition.
For example, let’s take a look at the following expression.
8x + 2y - 3
This expression contains three terms; 8x, 2y, and 3. The first two terms consist of both numbers (8 and 2) and variables (x and y).
Expressions that include variables, coefficients, and sometimes constants, are also referred to as polynomials.
Simplifying expressions is crucial because it lays the groundwork for grasping how to solve them. Expressions can be expressed in intricate ways, and without simplifying them, anyone will have a difficult time attempting to solve them, with more opportunity for a mistake.
Obviously, all expressions will differ regarding how they are simplified based on what terms they contain, but there are common steps that can be applied to all rational expressions of real numbers, whether they are logarithms, square roots, etc.
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 between 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 on the outside with the one inside.
Exponents. Where possible, use the exponent principles to simplify the terms that have exponents.
Multiplication and Division. If the equation necessitates it, use multiplication or division rules to simplify like terms that apply.
Addition and subtraction. Then, use addition or subtraction the resulting terms in the equation.
Rewrite. Ensure that there are no additional like terms that require simplification, and rewrite the simplified equation.
The Properties For Simplifying Algebraic Expressions
In addition to the PEMDAS sequence, there are a few more rules you should be informed of when working with algebraic expressions.
You can only simplify 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 equation 8x + 2x can be simplified to 10x by adding coefficients 8 and 2 and retaining the variable x as it is.
Parentheses that contain another expression directly outside of them need to use the distributive property. The distributive property gives you the ability to to simplify terms on the outside of parentheses by distributing them to the terms inside, as shown here: a(b+c) = ab + ac.
An extension of the distributive property is known as the principle of multiplication. When two stand-alone expressions within parentheses are multiplied, the distribution rule applies, and every individual term will will require multiplication by the other terms, making each set of equations, common factors of one another. Such as is the case here: (a + b)(c + d) = a(c + d) + b(c + d).
A negative sign right outside of an expression in parentheses denotes that the negative expression must also need to be distributed, changing the signs of the terms on the inside of the parentheses. As is the case in this example: -(8x + 2) will turn into -8x - 2.
Similarly, a plus sign right outside the parentheses denotes that it will be distributed to the terms inside. However, this means that you should eliminate 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 prior principles were simple enough to follow as they only dealt with principles that impact simple terms with numbers and variables. Still, there are a few other rules that you must implement when working with expressions with exponents.
Here, we will discuss the laws of exponents. Eight principles influence how we utilize exponentials, 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 a 1 exponent will not change in value. Or a1 = a.
Product Rule. When two terms with equivalent variables are multiplied by each other, their product will add their exponents. This is written as am × an = am+n
Quotient Rule. When two terms with the same variables are divided by each other, their quotient will subtract their applicable exponents. This is seen 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 end up being the 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 needs to 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.
Simplifying Expressions with the Distributive Property
The distributive property is the property that states that any term multiplied by an expression within parentheses should be multiplied by all of the expressions on the inside. 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 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 some 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 separately by their numerators and denominators.
Laws of exponents. This states 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 should be expressed in the expression. Use the PEMDAS rule and be sure that no two terms possess matching variables.
These are the same rules that you can apply when simplifying any real numbers, whether they are binomials, decimals, square roots, logarithms, linear equations, or quadratic equations.
Practice Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this case, the rules that need to be noted first are PEMDAS and the distributive property. The distributive property will distribute 4 to all the expressions inside of the parentheses, while PEMDAS will dictate the order of simplification.
As a result of the distributive property, the term outside 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 the same variables, and each term should be in its most simplified form.
28x + 28 - 3y
Rearrange the equation this way:
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 needs the distributive property. In this scenario, the term y/4 should be distributed within 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 attached to them. Since we know from PEMDAS that fractions will 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 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 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 the same 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 obey the exponential rule, the distributive property, and PEMDAS rules and the concept of multiplication of algebraic expressions. Finally, ensure 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 vastly different, but, they can be combined the same process due to the fact that you must first simplify expressions before you solve them.
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