Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can appear to be scary for new pupils in their early years of college or even in high school.
Nevertheless, understanding how to handle these equations is essential because it is foundational knowledge that will help them eventually be able to solve higher math and advanced problems across different industries.
This article will discuss everything you need to know simplifying expressions. We’ll review the laws of simplifying expressions and then validate what we've learned through some sample questions.
How Does Simplifying Expressions Work?
Before you can be taught how to simplify them, you must learn what expressions are in the first place.
In arithmetics, expressions are descriptions that have a minimum of two terms. These terms can include variables, numbers, or both and can be linked through addition or subtraction.
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 include 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 essential because it paves the way for learning how to solve them. Expressions can be expressed in complicated ways, and without simplifying them, anyone will have a difficult time trying to solve them, with more possibility for a mistake.
Obviously, each expression vary regarding how they are simplified based on what terms they incorporate, but there are general steps that are applicable to all rational expressions of real numbers, whether they are logarithms, square roots, etc.
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. Simplify equations between the parentheses first by using addition or subtracting. If there are terms right outside the parentheses, use the distributive property to apply multiplication the term on the outside with the one on the inside.
Exponents. Where possible, use the exponent properties to simplify the terms that include exponents.
Multiplication and Division. If the equation requires it, use multiplication or division rules to simplify like terms that are applicable.
Addition and subtraction. Then, add or subtract the remaining terms in the equation.
Rewrite. Ensure that there are no additional like terms that require simplification, then rewrite the simplified equation.
Here are the Properties For Simplifying Algebraic Expressions
Along with the PEMDAS rule, there are a few more properties you must be informed of when dealing with algebraic expressions.
You can only simplify terms with common variables. When adding these terms, add the coefficient numbers and maintain 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 leaving the x as it is.
Parentheses that contain another expression outside of them need to utilize 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, for example: 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 distribution principle kicks in, and all individual term will need to be multiplied by the other terms, making 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 right outside of an expression in parentheses means that the negative expression should also need to be distributed, changing the signs of the terms inside the parentheses. As is the case in this example: -(8x + 2) will turn into -8x - 2.
Similarly, a plus sign right outside the parentheses means that it will have distribution applied to the terms on the inside. Despite that, this means that you are able to 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 rules were simple enough to follow as they only dealt with properties that impact simple terms with variables and numbers. Still, there are a few other rules that you need to apply when dealing with exponents and expressions.
In this section, we will discuss the laws of exponents. 8 principles affect how we deal with exponentials, that includes 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 the same 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 matching variables are divided by each other, their quotient will subtract their two respective exponents. This is seen 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 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 different variables needs to be applied to the appropriate 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.
Simplifying Expressions with the Distributive Property
The distributive property is the principle that says that any term multiplied by an expression on the inside of a parentheses needs be multiplied by all of the expressions 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 result is 6x + 10.
How to Simplify Expressions with Fractions
Certain expressions contain fractions, and just as with exponents, expressions with fractions also have multiple rules that you must follow.
When an expression has 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 one at a time by their denominators and numerators.
Laws of exponents. This tells 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 state should be included in the expression. Use the PEMDAS principle and ensure that no two terms share matching variables.
These are the exact properties that you can apply when simplifying any real numbers, whether they are binomials, decimals, square roots, 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 rules that should be noted first are the distributive property and the PEMDAS rule. The distributive property will distribute 4 to all the expressions inside the parentheses, while PEMDAS will govern 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 all the terms with matching 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 first in order should be expressions on the inside of parentheses, and in this scenario, that expression also needs the distributive property. In this scenario, the term y/4 must be distributed to the two terms on the inside of the parentheses, as seen in this example.
1/3x + y/4(5x) + y/4(2)
Here, let’s set aside the first term for now and simplify the terms with factors associated with them. Because we know from PEMDAS that fractions require multiplication of their denominators and numerators individually, we will then have:
y/4 * 5x/1
The expression 5x/1 is used for simplicity 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 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, which means that we’ll have to add the exponents of two exponential expressions with similar variables multiplied together and multiply their coefficients. This gives us:
24x3 + 4x2 + 18xy + 3y
Due to the fact that there are no more like terms to be simplified, this becomes our final answer.
Simplifying Expressions FAQs
What should I remember when simplifying expressions?
When simplifying algebraic expressions, bear in mind that you are required to obey PEMDAS, the exponential rule, and the distributive property rules as well as the rule of multiplication of algebraic expressions. Ultimately, make sure that every term on your expression is in its most simplified form.
What is the difference between solving an equation and simplifying an expression?
Solving and simplifying expressions are very different, but, they can be incorporated into the same process the same process since you first need to simplify expressions before you begin solving them.
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