Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can appear to be scary for new students in their early years of high school or college.
Nevertheless, learning how to handle these equations is critical because it is foundational knowledge that will help them navigate higher math and complicated problems across various industries.
This article will discuss everything you need to master simplifying expressions. We’ll review the principles of simplifying expressions and then verify what we've learned through some sample problems.
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 at least two terms. These terms can include variables, numbers, or both and can be linked through subtraction or addition.
As an example, let’s go over the following expression.
8x + 2y - 3
This expression includes three terms; 8x, 2y, and 3. The first two include both numbers (8 and 2) and variables (x and y).
Expressions consisting of variables, coefficients, and sometimes constants, are also referred to as polynomials.
Simplifying expressions is essential because it lays the groundwork for understanding how to solve them. Expressions can be written in complicated ways, and without simplifying them, anyone will have a hard time trying to solve them, with more chance for solving them incorrectly.
Obviously, each expression differ concerning how they're simplified depending on what terms they include, 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, an abbreviation for parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule declares the order of operations for expressions.
Parentheses. Solve equations between 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 outside with the one inside.
Exponents. Where workable, use the exponent properties to simplify the terms that contain exponents.
Multiplication and Division. If the equation calls for it, utilize the multiplication and division principles to simplify like terms that are applicable.
Addition and subtraction. Lastly, use addition or subtraction the simplified terms of the equation.
Rewrite. Make sure that there are no additional like terms to simplify, then rewrite the simplified equation.
Here are the Rules For Simplifying Algebraic Expressions
In addition to the PEMDAS principle, there are a few more principles you must be aware of when simplifying algebraic expressions.
You can only simplify 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 adding coefficients 8 and 2 and retaining the variable x as it is.
Parentheses that contain another expression on the outside of them need to apply 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 on the inside, for example: a(b+c) = ab + ac.
An extension of the distributive property is referred to as the principle of multiplication. When two stand-alone expressions within parentheses are multiplied, the distributive rule 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 must 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 right outside the parentheses means that it will have distribution applied to the terms on the inside. Despite that, this means that you can eliminate the parentheses and write the expression as is owing to the fact that the plus sign doesn’t alter anything when distributed.
How to Simplify Expressions with Exponents
The previous principles were simple enough to implement as they only dealt with properties that affect simple terms with numbers and variables. However, there are a few other rules that you must apply when dealing with expressions with exponents.
Next, we will review the properties of exponents. 8 rules influence how we deal with exponents, 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 a 1 exponent doesn'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 exponents. This is written as am × an = am+n
Quotient Rule. When two terms with the same variables are divided, their quotient subtracts their respective exponents. This is seen as the formula am/an = am-n.
Negative Exponents Rule. Any term with a negative exponent is equivalent 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 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 possess different variables should be applied to the appropriate 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.
How to Simplify Expressions with the Distributive Property
The distributive property is the principle that states that any term multiplied by an expression on the inside of a parentheses must be multiplied by all of the expressions on the inside. Let’s witness 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 expression then becomes 6x + 10.
Simplifying Expressions with Fractions
Certain expressions can consist of fractions, and just as with exponents, expressions with fractions also have multiple rules that you need to 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 separately by their denominators and numerators.
Laws of exponents. This tells us that fractions will usually be the power of the quotient rule, which will subtract the exponents of the numerators and denominators.
Simplification. Only fractions at their lowest state should be included in the expression. Use the PEMDAS rule and ensure that no two terms have the same variables.
These are the exact principles that you can apply when simplifying any real numbers, whether they are binomials, decimals, square roots, quadratic equations, logarithms, or linear equations.
Sample Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
Here, the rules that need to be noted first are the distributive property and the PEMDAS rule. The distributive property will distribute 4 to all other expressions on the inside of the parentheses, while PEMDAS will govern the order of simplification.
As a result of the distributive property, the term outside 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, you should add the terms with matching variables, and each term should be in its lowest 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 within parentheses, and in this case, that expression also necessitates the distributive property. In this scenario, the term y/4 must be distributed within 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 attached to them. Remember 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 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 utilized to distribute every 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 similar variables multiplied together and multiply their coefficients. This gives us:
24x3 + 4x2 + 18xy + 3y
Because there are no other like terms to simplify, this becomes our final answer.
Simplifying Expressions FAQs
What should I keep in mind when simplifying expressions?
When simplifying algebraic expressions, remember that you must obey the exponential rule, the distributive property, and PEMDAS rules in addition to the rule of multiplication of algebraic expressions. Ultimately, make sure that every term on your expression is in its lowest form.
How does solving equations differ from simplifying expressions?
Simplifying and solving equations are quite different, but, they can be part of the same process the same process because you first need to simplify expressions before you begin solving them.
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