Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can appear to be challenging for beginner pupils in their primary years of college or even in high school.
Nevertheless, grasping how to handle these equations is important because it is basic knowledge that will help them navigate higher arithmetics and advanced problems across different industries.
This article will share everything you need to learn simplifying expressions. We’ll learn the laws of simplifying expressions and then test our skills via some practice problems.
How Does Simplifying Expressions Work?
Before you can be taught how to simplify them, you must understand what expressions are in the first place.
In mathematics, expressions are descriptions that have at least two terms. These terms can combine numbers, variables, or both and can be linked through addition or subtraction.
To give an example, let’s take a look at the following expression.
8x + 2y - 3
This expression combines 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 called polynomials.
Simplifying expressions is crucial because it opens up the possibility of learning how to solve them. Expressions can be expressed in intricate ways, and without simplification, everyone will have a difficult time trying to solve them, with more possibility for a mistake.
Undoubtedly, every expression be different concerning how they're simplified depending on what terms they include, but there are typical steps that can be applied to all rational expressions of real numbers, whether they are logarithms, square roots, etc.
These steps are called the PEMDAS rule, an abbreviation 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 applying subtraction. If there are terms just outside the parentheses, use the distributive property to multiply the term outside with the one inside.
Exponents. Where feasible, use the exponent rules to simplify the terms that have exponents.
Multiplication and Division. If the equation calls for it, utilize the multiplication and division principles to simplify like terms that apply.
Addition and subtraction. Then, add or subtract the simplified terms in the equation.
Rewrite. Make sure that there are no remaining like terms that need to be simplified, and rewrite the simplified equation.
Here are the Properties For Simplifying Algebraic Expressions
Beyond the PEMDAS rule, there are a few more properties you must be aware of when dealing with algebraic expressions.
You can only simplify terms with common variables. When applying addition to these terms, add the coefficient numbers and leave 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 retaining the variable x as it is.
Parentheses containing another expression on the outside of them need to apply the distributive property. The distributive property prompts you to simplify terms outside of parentheses by distributing them to the terms inside, for example: a(b+c) = ab + ac.
An extension of the distributive property is referred to as the principle of multiplication. When two distinct expressions within parentheses are multiplied, the distribution property kicks in, and each unique term will need to be multiplied by the other terms, making 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 directly outside of an expression in parentheses denotes that the negative expression must also need to be distributed, changing the signs of the terms inside the parentheses. Like 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 on the inside. However, this means that you can eliminate the parentheses and write the expression as is because the plus sign doesn’t change anything when distributed.
How to Simplify Expressions with Exponents
The prior properties were easy enough to implement as they only applied to rules that affect simple terms with numbers and variables. However, there are more rules that you have to implement when dealing with expressions with exponents.
Next, we will discuss the principles of exponents. Eight properties impact how we process exponentials, that includes the following:
Zero Exponent Rule. This property states that any term with a 0 exponent is equal to 1. Or a0 = 1.
Identity Exponent Rule. Any term with the exponent of 1 won'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 exponents. This is written as am × an = am+n
Quotient Rule. When two terms with matching variables are divided by each other, their quotient will subtract their 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 already with an exponent, the term will result in 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 different variables will 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 take 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 within parentheses needs be multiplied by all of the expressions on the inside. Let’s watch 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 resulting expression is 6x + 10.
Simplifying Expressions with Fractions
Certain expressions contain fractions, and just like 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. Refer to the PEMDAS property and be sure 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 square roots, binomials, decimals, linear equations, quadratic equations, and even logarithms.
Practice Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this case, the rules that should be noted first are the PEMDAS and the distributive property. The distributive property will distribute 4 to all the expressions on the inside of the parentheses, while PEMDAS will decide on the order of simplification.
Because of the distributive property, the term on the 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, 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 the order should start with expressions inside parentheses, and in this case, that expression also needs the distributive property. Here, the term y/4 must be distributed amongst 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 put aside the first term for now and simplify the terms with factors assigned to them. Because we know from PEMDAS that fractions will need to multiply their denominators and numerators individually, we will then have:
y/4 * 5x/1
The expression 5x/1 is used for simplicity 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 utilized 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, 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 other 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, keep in mind that you have to obey the distributive property, PEMDAS, and the exponential rule rules as well as the rule of multiplication of algebraic expressions. Ultimately, ensure 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 very 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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