Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can appear to be scary for beginner learners in their first years of high school or college.
However, understanding how to deal with these equations is important because it is primary knowledge that will help them navigate higher arithmetics and complex problems across multiple industries.
This article will go over everything you need to master simplifying expressions. We’ll learn the laws of simplifying expressions and then test our comprehension with some sample problems.
How Do You Simplify Expressions?
Before learning how to simplify them, you must learn what expressions are to begin with.
In mathematics, expressions are descriptions that have no less than two terms. These terms can combine variables, numbers, or both and can be connected through subtraction or addition.
To give an example, let’s go over the following expression.
8x + 2y - 3
This expression combines three terms; 8x, 2y, and 3. The first two contain 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 crucial because it lays the groundwork for understanding how to solve them. Expressions can be expressed in complicated ways, and without simplification, everyone will have a hard time attempting to solve them, with more possibility for a mistake.
Obviously, every expression differ in how they're simplified based on what terms they include, but there are typical steps that apply to all rational expressions of real numbers, regardless of whether they are logarithms, square roots, etc.
These steps are called the PEMDAS rule, or parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule shows us 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 on the outside with the one inside.
Exponents. Where possible, use the exponent principles to simplify the terms that have exponents.
Multiplication and Division. If the equation calls for it, use multiplication or division rules to simplify like terms that are applicable.
Addition and subtraction. Lastly, add or subtract the resulting terms in the equation.
Rewrite. Ensure that there are no more like terms that require simplification, and rewrite the simplified equation.
Here are the Requirements For Simplifying Algebraic Expressions
Along with the PEMDAS principle, there are a few more properties you should be aware of when simplifying algebraic expressions.
You can only apply simplification to terms with common variables. When applying addition to these terms, add the coefficient numbers and keep 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 x as it is.
Parentheses that include another expression on the outside of them need to apply the distributive property. The distributive property allows 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 referred to as the property of multiplication. When two separate expressions within parentheses are multiplied, the distributive rule applies, and every separate term will need to be multiplied by the other terms, making each set of equations, common factors of one another. Like in this example: (a + b)(c + d) = a(c + d) + b(c + d).
A negative sign outside an expression in parentheses denotes that the negative expression will also need to have distribution applied, 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 on the outside of the parentheses will mean that it will be distributed to the terms on the inside. Despite that, this means that you should remove 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 simple enough to use as they only applied to principles that affect simple terms with variables and numbers. However, there are more rules that you need to implement when dealing with expressions with exponents.
In this section, we will discuss the laws of exponents. Eight properties impact how we utilize exponentials, that includes the following:
Zero Exponent Rule. This property states that any term with a 0 exponent equals 1. Or a0 = 1.
Identity Exponent Rule. Any term with a 1 exponent doesn't alter the value. Or a1 = a.
Product Rule. When two terms with equivalent variables are multiplied, their product will add their exponents. This is written as am × an = am+n
Quotient Rule. When two terms with matching variables are divided, their quotient applies subtraction to their applicable 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 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 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 denominator and numerator will acquire the exponent given, (a/b)m = am/bm.
Simplifying Expressions with the Distributive Property
The distributive property is the rule that denotes 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 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.
Simplifying Expressions with Fractions
Certain expressions contain fractions, and just like with exponents, expressions with fractions also have multiple rules that you have to follow.
When an expression contains 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 numerators and denominators.
Laws of exponents. This tells us that fractions will more likely be the power of the quotient rule, which will subtract the exponents of the denominators and numerators.
Simplification. Only fractions at their lowest form should be included in the expression. Apply the PEMDAS rule and make sure that no two terms have matching variables.
These are the same principles that you can apply when simplifying any real numbers, whether they are decimals, square roots, binomials, 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 rules that need to be noted first are PEMDAS and the distributive property. The distributive property will distribute 4 to all other expressions inside the parentheses, while PEMDAS will decide on the order of simplification.
Due to the distributive property, the term outside of 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, remember to add the terms with matching 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 expresses that the the order should start with expressions inside parentheses, and in this scenario, that expression also needs the distributive property. In this example, the term y/4 should be distributed to the two terms on the inside of 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 require multiplication of their denominators and numerators individually, we will then have:
y/4 * 5x/1
The expression 5x/1 is used for simplicity 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 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, 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
Since there are no more like terms to simplify, this becomes our final answer.
Simplifying Expressions FAQs
What should I bear in mind when simplifying expressions?
When simplifying algebraic expressions, remember that you must obey the distributive property, PEMDAS, and the exponential rule rules and the principle of multiplication of algebraic expressions. Finally, ensure that every term on your expression is in its lowest form.
How does solving equations differ from simplifying expressions?
Solving and simplifying expressions are vastly different, although, they can be part of the same process the same process due to the fact that you must first simplify expressions before solving them.
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