Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be intimidating for budding students in their first years of high school or college.
However, grasping how to process these equations is essential because it is primary information that will help them eventually be able to solve higher mathematics and complex problems across multiple industries.
This article will discuss everything you should review to know simplifying expressions. We’ll learn the proponents of simplifying expressions and then test our comprehension through some practice questions.
How Does Simplifying Expressions Work?
Before learning how to simplify them, you must learn what expressions are at their core.
In arithmetics, expressions are descriptions that have no less than two terms. These terms can contain variables, numbers, or both and can be linked through subtraction or addition.
As an example, let’s review the following expression.
8x + 2y - 3
This expression combines three terms; 8x, 2y, and 3. The first two terms include both numbers (8 and 2) and variables (x and y).
Expressions containing variables, coefficients, and sometimes constants, are also known as polynomials.
Simplifying expressions is crucial because it paves the way for understanding how to solve them. Expressions can be expressed in intricate ways, and without simplification, everyone will have a difficult time attempting to solve them, with more possibility for solving them incorrectly.
Undoubtedly, each expression differ concerning how they're simplified based on what terms they incorporate, but there are common 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, short for parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule declares the order of operations for expressions.
Parentheses. Resolve equations within the parentheses first by applying addition or applying subtraction. If there are terms right outside the parentheses, use the distributive property to multiply the term on the outside with the one on the inside.
Exponents. Where possible, use the exponent principles 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, add or subtract the remaining terms of the equation.
Rewrite. Make sure that there are no more like terms that need to be simplified, then rewrite the simplified equation.
The Rules For Simplifying Algebraic Expressions
Along with the PEMDAS sequence, there are a few additional principles you should be aware of when working with algebraic expressions.
You can only apply simplification to 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 equation 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 that include another expression directly outside of them need to utilize the distributive property. The distributive property gives you the ability to 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 called the concept of multiplication. When two stand-alone expressions within parentheses are multiplied, the distribution property is applied, and every separate term will will require multiplication by the other terms, making 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 denotes that the negative expression should 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.
Likewise, a plus sign right outside the parentheses will mean that it will be distributed to the terms on the inside. Despite that, this means that you should eliminate the parentheses and write the expression as is because the plus sign doesn’t alter anything when distributed.
How to Simplify Expressions with Exponents
The prior principles were straight-forward enough to use as they only dealt with rules that impact simple terms with variables and numbers. However, there are more rules that you have to implement when working with exponents and expressions.
Next, we will discuss the properties of exponents. 8 rules influence how we utilize exponentials, that includes the following:
Zero Exponent Rule. This rule states that any term with the exponent of 0 equals 1. Or a0 = 1.
Identity Exponent Rule. Any term with the exponent of 1 doesn'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, their quotient will subtract their applicable exponents. This is expressed in the formula am/an = am-n.
Negative Exponents Rule. Any term with a negative exponent equals 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 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 numerator and denominator will assume the exponent given, (a/b)m = am/bm.
How to Simplify Expressions with the Distributive Property
The distributive property is the rule that states that any term multiplied by an expression within parentheses should be multiplied by all of the expressions within. Let’s watch 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 resulting expression is 6x + 10.
How to Simplify Expressions with Fractions
Certain expressions can consist of fractions, and just like with exponents, expressions with fractions also have several 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 one at a time by their denominators and numerators.
Laws of exponents. This tells us that fractions will usually be the power of the quotient rule, which will subtract the exponents of the numerators and denominators.
Simplification. Only fractions at their lowest form should be included in the expression. Use the PEMDAS principle and be sure that no two terms contain matching variables.
These are the exact rules that you can apply when simplifying any real numbers, whether they are square roots, binomials, decimals, quadratic equations, logarithms, or linear equations.
Practice Examples for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
Here, the principles that need to be noted first are the PEMDAS and the distributive property. The distributive property will distribute 4 to the expressions inside of the parentheses, while PEMDAS will govern the order of simplification.
Due to 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 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 states that the the order should start with expressions within parentheses, and in this example, that expression also requires the distributive property. Here, the term y/4 will need to be distributed to 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 now and simplify the terms with factors associated with them. Because we know from PEMDAS that fractions require multiplication of their numerators and denominators 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 used to distribute every term 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, which tells us 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 more 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 have to 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.
What is the difference between solving an equation and simplifying an expression?
Solving equations and simplifying expressions are vastly different, however, they can be combined the same process due to the fact that you must first simplify expressions before solving them.
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