Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions are one of the most scary for budding learners in their early years of college or even in high school.
However, learning how to handle these equations is important because it is primary knowledge that will help them navigate higher mathematics and complicated problems across multiple industries.
This article will discuss everything you need to learn simplifying expressions. We’ll review the proponents of simplifying expressions and then validate our skills through some practice problems.
How Do I Simplify an Expression?
Before learning how to simplify expressions, you must understand what expressions are at their core.
In mathematics, expressions are descriptions that have at least two terms. These terms can contain variables, numbers, or both and can be connected through addition or subtraction.
For example, let’s go over the following expression.
8x + 2y - 3
This expression includes three terms; 8x, 2y, and 3. The first two consist of both numbers (8 and 2) and variables (x and y).
Expressions that incorporate coefficients, variables, and occasionally constants, are also known as polynomials.
Simplifying expressions is crucial because it opens up the possibility of understanding how to solve them. Expressions can be expressed in intricate ways, and without simplifying them, anyone will have a tough time trying to solve them, with more possibility for solving them incorrectly.
Undoubtedly, every expression vary concerning how they're simplified based on what terms they include, but there are general steps that are applicable to all rational expressions of real numbers, regardless of whether they are logarithms, square roots, etc.
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. Solve equations inside the parentheses first by applying addition or using subtraction. If there are terms just outside the parentheses, use the distributive property to multiply the term on the outside with the one on the inside.
Exponents. Where feasible, 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. Ensure that there are no more like terms that need to be simplified, then rewrite the simplified equation.
Here are the Properties For Simplifying Algebraic Expressions
Along with the PEMDAS sequence, there are a few more properties you need to be informed of when simplifying algebraic expressions.
You can only apply simplification to terms with common variables. When adding 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 x as it is.
Parentheses that contain another expression directly outside of them need to apply the distributive property. The distributive property gives you the ability to 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 known as the principle of multiplication. When two separate expressions within parentheses are multiplied, the distributive principle is applied, and all individual term will will require multiplication by the other terms, resulting in each set of equations, common factors of each other. Such as is the case here: (a + b)(c + d) = a(c + d) + b(c + d).
A negative sign right outside of an expression in parentheses means 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 means that it will be distributed to the terms inside. But, 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 rules were easy enough to use as they only applied to rules that affect simple terms with variables and numbers. Still, there are more rules that you must apply when dealing with exponents and expressions.
Next, we will review the laws of exponents. Eight rules impact how we deal with exponents, those 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 a 1 exponent won't alter the 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, 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 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 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 have 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 take the exponent given, (a/b)m = am/bm.
Simplifying Expressions with the Distributive Property
The distributive property is the principle that shows us that any term multiplied by an expression within parentheses should be multiplied by all of the expressions within. Let’s watch 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 contain fractions, and just like with exponents, expressions with fractions also have some rules that you need to follow.
When an expression consist of fractions, here's 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 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 state should be included in the expression. Use the PEMDAS rule and be sure that no two terms share the same variables.
These are the exact properties that you can apply when simplifying any real numbers, whether they are binomials, decimals, square roots, quadratic equations, logarithms, or linear equations.
Practice Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this example, the properties that must be noted first are the distributive property and the PEMDAS rule. The distributive property will distribute 4 to all the expressions inside 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 each term on the inside.
4(2x) + 4(5x) + 4(7) - 3y
8x + 20x + 28 - 3y
When simplifying equations, remember to add all the terms with the same 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 expresses that the you should begin with expressions within parentheses, and in this example, that expression also requires the distributive property. In this scenario, the term y/4 will need to be distributed within the two terms inside the parentheses, as seen here.
1/3x + y/4(5x) + y/4(2)
Here, let’s put aside the first term for the moment and simplify the terms with factors assigned to 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 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 used 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
Because there are no more like terms to be simplified, this becomes our final answer.
Simplifying Expressions FAQs
What should I remember when simplifying expressions?
When simplifying algebraic expressions, remember that you have to follow the distributive property, PEMDAS, and the exponential rule rules in addition to the concept of multiplication of algebraic expressions. In the end, make sure that every term on your expression is in its lowest form.
How does solving equations differ from simplifying expressions?
Solving equations and simplifying expressions are vastly different, although, they can be part of the same process the same process because you have to simplify expressions before solving them.
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