Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be scary for beginner pupils in their early years of college or even in high school.
Nevertheless, understanding how to handle these equations is essential because it is primary knowledge that will help them eventually be able to solve higher math and complex problems across multiple industries.
This article will discuss everything you should review to know simplifying expressions. We’ll review the proponents of simplifying expressions and then verify our skills through some practice problems.
How Do You Simplify Expressions?
Before you can learn how to simplify expressions, you must learn what expressions are at their core.
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 take a look at the following expression.
8x + 2y - 3
This expression contains three terms; 8x, 2y, and 3. The first two terms consist of both numbers (8 and 2) and variables (x and y).
Expressions that include coefficients, variables, and occasionally constants, are also referred to as polynomials.
Simplifying expressions is essential because it opens up the possibility of understanding how to solve them. Expressions can be written in intricate ways, and without simplification, everyone will have a tough time attempting to solve them, with more opportunity for a mistake.
Of course, every expression differ in how they're 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 refered to as the PEMDAS rule, an abbreviation 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 using addition 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 rules to simplify the terms that include exponents.
Multiplication and Division. If the equation calls for it, utilize multiplication and division to simplify like terms that apply.
Addition and subtraction. Then, use addition or subtraction the simplified terms in the equation.
Rewrite. Make sure that there are no remaining like terms to simplify, and then rewrite the simplified equation.
Here are the Requirements For Simplifying Algebraic Expressions
Along with the PEMDAS principle, there are a few more principles you need to be aware of when dealing with algebraic expressions.
You can only simplify terms with common variables. When adding these terms, add the coefficient numbers and keep the variables as [[is|they are]-70. For example, the equation 8x + 2x can be simplified to 10x by adding coefficients 8 and 2 and keeping the variable x as it is.
Parentheses that include another expression outside of them need to utilize the distributive property. The distributive property allows you to simplify terms outside of parentheses by distributing them to the terms inside, or as follows: a(b+c) = ab + ac.
An extension of the distributive property is referred to as the property of multiplication. When two distinct expressions within parentheses are multiplied, the distributive rule kicks in, and all unique term will need 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 outside 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. Like in this example: -(8x + 2) will turn into -8x - 2.
Likewise, a plus sign right outside the parentheses means that it will be distributed to the terms inside. However, 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 principles were straight-forward enough to use as they only dealt with properties that affect simple terms with numbers and variables. Still, there are additional rules that you have to follow when working with expressions with exponents.
Next, we will discuss the properties of exponents. 8 rules affect how we utilize exponentials, which are 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 the exponent of 1 doesn't change in value. Or a1 = a.
Product Rule. When two terms with equivalent 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 the same variables are divided by each other, their quotient will subtract their two respective exponents. This is expressed in 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 that already has 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 have different variables should be applied to the required variables, or (ab)m = am * bm.
Power of a Quotient Rule. In fractional exponents, both the denominator and numerator will take the exponent given, (a/b)m = am/bm.
Simplifying 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 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 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 contains fractions, here's what to remember.
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 shows us that fractions will more likely 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 expressed in the expression. Apply the PEMDAS rule and make sure that no two terms share 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.
Sample Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this example, 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 govern the order of simplification.
As a result of the distributive property, the term outside the parentheses will be multiplied by the terms 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 every term should be in its most simplified form.
28x + 28 - 3y
Rearrange the equation like this:
28x - 3y + 28
Example 2
Simplify the expression 1/3x + y/4(5x + 2)
The PEMDAS rule expresses that the the order should start with expressions on the inside of parentheses, and in this scenario, that expression also requires the distributive property. In this scenario, the term y/4 will need to be distributed within the two terms within the parentheses, as follows.
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 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 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 all terms 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, 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 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 as well as the concept of multiplication of algebraic expressions. Finally, make sure that every term on your expression is in its most simplified form.
How does solving equations differ from simplifying expressions?
Simplifying and solving equations are very different, although, they can be incorporated into the same process the same process because you have to simplify expressions before solving them.
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