Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be challenging for beginner learners in their first years of high school or college.
Still, understanding how to handle these equations is critical because it is primary information that will help them move on to higher math and complex problems across various industries.
This article will discuss everything you must have to learn simplifying expressions. We’ll review the principles of simplifying expressions and then verify what we've learned via some practice questions.
How Do You Simplify Expressions?
Before you can be taught how to simplify expressions, you must understand what expressions are in the first place.
In arithmetics, expressions are descriptions that have no less than two terms. These terms can contain numbers, variables, or both and can be linked through subtraction or addition.
For example, let’s take a look at the following expression.
8x + 2y - 3
This expression includes three terms; 8x, 2y, and 3. The first two contain both numbers (8 and 2) and variables (x and y).
Expressions consisting of variables, coefficients, and sometimes constants, are also known as polynomials.
Simplifying expressions is important because it opens up the possibility of learning how to solve them. Expressions can be expressed in complicated ways, and without simplifying them, everyone will have a hard time attempting to solve them, with more chance for error.
Of course, each expression differ in how they are simplified depending on what terms they contain, but there are common steps that are applicable to all rational expressions of real numbers, 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 states that the order of operations for expressions.
Parentheses. Resolve equations within the parentheses first by adding or subtracting. If there are terms right outside the parentheses, use the distributive property to apply multiplication the term outside with the one on the inside.
Exponents. Where workable, use the exponent properties to simplify the terms that include exponents.
Multiplication and Division. If the equation necessitates it, use multiplication and division to simplify like terms that are applicable.
Addition and subtraction. Finally, use addition or subtraction the remaining terms in the equation.
Rewrite. Ensure that there are no more like terms that require simplification, then rewrite the simplified equation.
Here are the Properties For Simplifying Algebraic Expressions
In addition to the PEMDAS principle, there are a few more principles you must be informed 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 applying addition to the coefficients 8 and 2 and leaving the variable x as it is.
Parentheses that contain another expression on the 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 on the inside, as shown here: a(b+c) = ab + ac.
An extension of the distributive property is known as the principle of multiplication. When two separate expressions within parentheses are multiplied, the distributive principle applies, and each separate term will will require multiplication by the other terms, resulting in each set of equations, common factors of each other. Like in this example: (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 should also need to have distribution applied, changing the signs of the terms inside the parentheses. For example: -(8x + 2) will turn into -8x - 2.
Similarly, a plus sign on the outside of the parentheses denotes that it will have distribution applied to the terms inside. However, this means that you should eliminate the parentheses and write the expression as is due to the fact that the plus sign doesn’t change anything when distributed.
How to Simplify Expressions with Exponents
The prior principles were easy enough to implement as they only dealt with principles that affect simple terms with numbers and variables. Despite that, there are a few other rules that you need to apply when dealing with expressions with exponents.
Next, we will review the laws of exponents. Eight rules affect how we process exponents, that includes the following:
Zero Exponent Rule. This rule states that any term with the exponent of 0 is equivalent to 1. Or a0 = 1.
Identity Exponent Rule. Any term with a 1 exponent won't alter the 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 matching variables are divided by each other, their quotient applies subtraction to their applicable exponents. This is expressed in 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 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 unique variables should 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.
How to Simplify Expressions with the Distributive Property
The distributive property is the property that says that any term multiplied by an expression on the inside of a parentheses needs be multiplied by all of the expressions on the inside. Let’s see 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 can consist of fractions, and just as with exponents, expressions with fractions also have several rules that you must follow.
When an expression consist of fractions, here is what to remember.
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 apply subtraction to the exponents of the denominators and numerators.
Simplification. Only fractions at their lowest form should be included in the expression. Use the PEMDAS principle and ensure that no two terms have the same variables.
These are the same rules that you can apply when simplifying any real numbers, whether they are decimals, square roots, binomials, quadratic equations, logarithms, or linear equations.
Practice Examples for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this case, the rules that must be noted first are the PEMDAS and the distributive property. The distributive property will distribute 4 to all other 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, you should 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 within parentheses, and in this example, that expression also needs the distributive property. In this scenario, the term y/4 must be distributed amongst the two terms inside the parentheses, as seen in this example.
1/3x + y/4(5x) + y/4(2)
Here, let’s set aside the first term for the moment and simplify the terms with factors assigned 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 since 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 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 the same variables multiplied together and multiply their coefficients. This gives us:
24x3 + 4x2 + 18xy + 3y
Since there are no remaining like terms to apply simplification to, this becomes our final answer.
Simplifying Expressions FAQs
What should I bear in mind when simplifying expressions?
When simplifying algebraic expressions, keep in mind that you must obey the exponential rule, the distributive property, and PEMDAS rules in addition to the rule of multiplication of algebraic expressions. Finally, make sure that every term on your expression is in its lowest form.
How are simplifying expressions and solving equations different?
Solving equations and simplifying expressions are quite different, however, they can be part of the same process the same process since you must first simplify expressions before solving them.
Let Grade Potential Help You Bone Up On Your Math
Simplifying algebraic equations is one of the most primary precalculus skills you must practice. Increasing your skill with simplification tactics and laws will pay dividends when you’re practicing sophisticated mathematics!
But these principles and laws can get challenging fast. Grade Potential is here to guide you, so don’t worry!
Grade Potential Arlington offers professional instructors that will get you where you need to be at your convenience. Our professional instructors will guide you applying mathematical properties in a straight-forward way to guide.
Contact us now!
