Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions are one of the most scary for beginner students in their first years of college or even in high school.
Still, learning how to handle these equations is essential because it is basic information that will help them move on to higher math and advanced problems across multiple industries.
This article will discuss everything you must have to know simplifying expressions. We’ll learn the proponents of simplifying expressions and then test our comprehension via some sample problems.
How Do I Simplify an Expression?
Before learning how to simplify them, you must learn what expressions are at their core.
In mathematics, expressions are descriptions that have at least two terms. These terms can include variables, numbers, or both and can be connected through addition or subtraction.
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 terms include both numbers (8 and 2) and variables (x and y).
Expressions that include variables, coefficients, and sometimes constants, are also called polynomials.
Simplifying expressions is crucial because it lays the groundwork for learning how to solve them. Expressions can be expressed in intricate ways, and without simplifying them, anyone will have a tough time attempting to solve them, with more chance for solving them incorrectly.
Obviously, all expressions will differ regarding how they are simplified depending on what terms they contain, but there are general 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 called the PEMDAS rule, short for parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule declares the order of operations for expressions.
Parentheses. Resolve equations between the parentheses first by adding or using subtraction. If there are terms just outside the parentheses, use the distributive property to apply multiplication the term on the outside with the one inside.
Exponents. Where possible, use the exponent properties to simplify the terms that contain exponents.
Multiplication and Division. If the equation calls for it, use the multiplication and division principles to simplify like terms that are applicable.
Addition and subtraction. Then, use addition or subtraction the simplified terms in the equation.
Rewrite. Make sure that there are no additional like terms to simplify, and rewrite the simplified equation.
Here are the Properties For Simplifying Algebraic Expressions
Beyond the PEMDAS principle, there are a few additional rules you need to 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 containing another expression directly outside of them need to use the distributive property. The distributive property prompts you 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 called the principle of multiplication. When two stand-alone expressions within parentheses are multiplied, the distribution rule applies, and each separate term will will require multiplication 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 directly outside of an expression in parentheses means 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 outside the parentheses will mean that it will have distribution applied to the terms inside. Despite that, this means that you are able to eliminate the parentheses and write the expression as is due to the fact that the plus sign doesn’t alter anything when distributed.
How to Simplify Expressions with Exponents
The previous properties were simple enough to implement as they only dealt with principles that affect simple terms with variables and numbers. However, there are a few other rules that you need to implement when working with exponents and expressions.
In this section, we will discuss the laws of exponents. 8 rules affect how we process exponents, those are the following:
Zero Exponent Rule. This rule states that any term with a 0 exponent equals 1. Or a0 = 1.
Identity Exponent Rule. Any term with the exponent of 1 will not 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 the same variables are divided by each other, their quotient applies subtraction to their respective 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 appropriate variables, or (ab)m = am * bm.
Power of a Quotient Rule. In fractional exponents, both the numerator and denominator will assume the exponent given, (a/b)m = am/bm.
How to Simplify Expressions with the Distributive Property
The distributive property is the principle that says that any term multiplied by an expression within parentheses should be multiplied by all of the expressions 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 result is 6x + 10.
Simplifying Expressions with Fractions
Certain expressions can consist of fractions, and just as with exponents, expressions with fractions also have several rules that you must 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 states that fractions will more likely 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 written in the expression. Refer to the PEMDAS property and be sure that no two terms contain matching variables.
These are the exact properties 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 principles that should be noted first are the PEMDAS and the distributive property. The distributive property will distribute 4 to the expressions on the inside of the parentheses, while PEMDAS will decide on the order of simplification.
As a result of the distributive property, the term 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 the terms with the same variables, and all 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 the order should start with expressions on the inside of parentheses, and in this example, that expression also necessitates the distributive property. Here, the term y/4 must be distributed to the two terms within the parentheses, as follows.
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 require multiplication of their numerators and denominators separately, 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 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 remaining like terms to apply simplification to, this becomes our final answer.
Simplifying Expressions FAQs
What should I keep in mind when simplifying expressions?
When simplifying algebraic expressions, remember that you have to obey 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 lowest form.
What is the difference between solving an equation and simplifying an expression?
Solving equations and simplifying expressions are quite different, but, they can be combined the same process because you have to simplify expressions before you begin solving them.
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