Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can appear to be challenging for beginner learners in their primary years of college or even in high school.
Still, grasping how to deal with these equations is critical because it is basic information that will help them navigate higher arithmetics and complicated problems across various industries.
This article will discuss everything you need to master simplifying expressions. We’ll review the laws of simplifying expressions and then validate what we've learned via some practice questions.
How Does Simplifying Expressions Work?
Before you can learn how to simplify expressions, you must learn what expressions are in the first place.
In mathematics, expressions are descriptions that have no less than two terms. These terms can include variables, numbers, or both and can be connected through subtraction or addition.
To give an example, let’s review the following expression.
8x + 2y - 3
This expression contains three terms; 8x, 2y, and 3. The first two consist of 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 essential because it lays the groundwork for learning how to solve them. Expressions can be written in complicated ways, and without simplifying them, everyone will have a difficult time attempting to solve them, with more possibility for a mistake.
Of course, all expressions will differ regarding how they are simplified depending 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 known as the PEMDAS rule, short for parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule shows us the order of operations for expressions.
Parentheses. Simplify equations inside the parentheses first by applying addition or applying subtraction. If there are terms just outside the parentheses, use the distributive property to multiply the term on the outside with the one on the inside.
Exponents. Where workable, use the exponent rules to simplify the terms that include exponents.
Multiplication and Division. If the equation requires it, utilize multiplication and division to simplify like terms that apply.
Addition and subtraction. Then, use addition or subtraction the remaining terms of the equation.
Rewrite. Ensure that there are no remaining like terms that need to be simplified, then rewrite the simplified equation.
The Requirements For Simplifying Algebraic Expressions
Along with the PEMDAS rule, there are a few additional principles you should be informed of when dealing with algebraic expressions.
You can only apply simplification to terms with common variables. When adding these terms, add the coefficient numbers and leave the variables as [[is|they are]-70. For example, the equation 8x + 2x can be simplified to 10x by applying addition to the coefficients 8 and 2 and keeping the variable x as it is.
Parentheses that include another expression 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, as shown here: a(b+c) = ab + ac.
An extension of the distributive property is known as the concept of multiplication. When two stand-alone expressions within parentheses are multiplied, the distributive property applies, and each individual term will have to be multiplied by the other terms, resulting in 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 directly outside of an expression in parentheses denotes that the negative expression must also need to have distribution applied, changing the signs of the terms on the inside of the parentheses. For example: -(8x + 2) will turn into -8x - 2.
Similarly, a plus sign right outside the parentheses means that it will be distributed to the terms inside. But, this means that you are able to eliminate the parentheses and write the expression as is because the plus sign doesn’t change anything when distributed.
How to Simplify Expressions with Exponents
The previous rules were easy enough to use as they only dealt with rules that affect simple terms with variables and numbers. Despite that, there are a few other rules that you have to apply when dealing with expressions with exponents.
Next, we will discuss the laws of exponents. 8 rules affect how we utilize exponentials, those are the following:
Zero Exponent Rule. This rule states that any term with the exponent of 0 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 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 applicable exponents. This is written as the formula am/an = am-n.
Negative Exponents Rule. Any term with a negative exponent equals 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 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 have different variables needs to be applied to the respective 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 rule that denotes that any term multiplied by an expression on the inside of a parentheses should be multiplied by all of the expressions on the inside. Let’s watch the distributive property applied 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 expression then becomes 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 need to follow.
When an expression includes fractions, here's what to keep in mind.
Distributive property. The distributive property a(b+c) = ab + ac, when applied to fractions, will multiply fractions separately by their denominators and numerators.
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 numerators and denominators.
Simplification. Only fractions at their lowest form should be written in the expression. Apply the PEMDAS rule and be sure that no two terms contain the same variables.
These are the same principles that you can apply when simplifying any real numbers, whether they are square roots, binomials, decimals, logarithms, linear equations, or quadratic equations.
Practice Examples 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 the distributive property and the PEMDAS rule. The distributive property will distribute 4 to all the expressions on the inside of 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 each term on the inside.
4(2x) + 4(5x) + 4(7) - 3y
8x + 20x + 28 - 3y
When simplifying equations, be sure to add the terms with matching 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 expresses that the you should begin with expressions on the inside of parentheses, and in this case, that expression also requires the distributive property. In this scenario, the term y/4 will need to be distributed to 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 associated with them. Remember we know from PEMDAS that fractions will need to multiply their denominators and numerators individually, 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 used to distribute every term 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, which means 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
Because there are no other like terms to be simplified, this becomes our final answer.
Simplifying Expressions FAQs
What should I bear in mind when simplifying expressions?
When simplifying algebraic expressions, remember that you must obey PEMDAS, the exponential rule, and the distributive property rules in addition to the principle of multiplication of algebraic expressions. In the end, make sure that every term on your expression is in its lowest form.
What is the difference between solving an equation and simplifying an expression?
Solving equations and simplifying expressions are vastly different, however, they can be part of the same process the same process due to the fact that you first need to simplify expressions before solving them.
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