Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can appear to be challenging for beginner learners in their first years of college or even in high school.
However, learning how to process these equations is important because it is foundational information that will help them navigate higher math and complicated problems across different industries.
This article will discuss everything you must have to master simplifying expressions. We’ll review the proponents of simplifying expressions and then validate our skills via some sample questions.
How Does Simplifying Expressions Work?
Before you can be taught how to simplify expressions, you must learn what expressions are in the first place.
In arithmetics, expressions are descriptions that have at least two terms. These terms can include numbers, variables, or both and can be connected 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 include both numbers (8 and 2) and variables (x and y).
Expressions that incorporate coefficients, variables, and occasionally constants, are also referred to as polynomials.
Simplifying expressions is important because it lays the groundwork for understanding how to solve them. Expressions can be written in complicated ways, and without simplifying them, you will have a hard time attempting to solve them, with more opportunity for solving them incorrectly.
Of course, each expression be different concerning how they are simplified based on what terms they include, but there are common steps that are applicable to all rational expressions of real numbers, regardless of whether they are logarithms, square roots, etc.
These steps are known as the PEMDAS rule, short for parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule declares the order of operations for expressions.
Parentheses. Simplify equations between 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 outside with the one on the inside.
Exponents. Where possible, use the exponent principles to simplify the terms that contain exponents.
Multiplication and Division. If the equation requires it, utilize multiplication or division rules to simplify like terms that apply.
Addition and subtraction. Then, add or subtract the remaining terms in the equation.
Rewrite. Make sure that there are no additional like terms to simplify, and then rewrite the simplified equation.
The Properties For Simplifying Algebraic Expressions
Along with the PEMDAS rule, there are a few more principles 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 equation 8x + 2x can be simplified to 10x by adding coefficients 8 and 2 and leaving the variable x as it is.
Parentheses that contain another expression outside of them need to use the distributive property. The distributive property gives you the ability to to simplify terms outside of parentheses by distributing them to the terms on the inside, for example: a(b+c) = ab + ac.
An extension of the distributive property is known as the principle of multiplication. When two stand-alone expressions within parentheses are multiplied, the distributive principle kicks in, and each individual term will will require multiplication 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 directly outside of an expression in parentheses indicates that the negative expression will also need to be distributed, changing the signs of the terms inside the parentheses. For example: -(8x + 2) will turn into -8x - 2.
Likewise, a plus sign right outside the parentheses means that it will have distribution applied to the terms inside. Despite that, this means that you are able to 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 previous principles were straight-forward enough to use as they only applied to properties that affect simple terms with variables and numbers. However, there are additional rules that you must follow when working with exponents and expressions.
Next, we will review the principles of exponents. Eight rules impact how we utilize exponents, which are the following:
Zero Exponent Rule. This rule states that any term with a 0 exponent equals 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 the same 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 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 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 that already has an exponent, the term will result in having a 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 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 rule that says that any term multiplied by an expression on the inside of a parentheses must be multiplied by all of the expressions inside. Let’s see 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 resulting expression is 6x + 10.
Simplifying Expressions with Fractions
Certain expressions can consist of fractions, and just like with exponents, expressions with fractions also have multiple rules that you need to follow.
When an expression has 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 states that fractions will usually 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 expressed in the expression. Apply the PEMDAS principle and make sure that no two terms contain matching variables.
These are the same principles that you can apply when simplifying any real numbers, whether they are binomials, decimals, square roots, linear equations, quadratic equations, and even logarithms.
Sample 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 distributive property and the PEMDAS rule. The distributive property will distribute 4 to the expressions inside the parentheses, while PEMDAS will decide on the order of simplification.
Due to the distributive property, the term on the outside of the parentheses will be multiplied by the terms 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 every 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 expresses that the the order should start with expressions inside parentheses, and in this case, that expression also requires the distributive property. In this example, the term y/4 will need to be distributed to the two terms on the inside of the parentheses, as seen here.
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. Since we know from PEMDAS that fractions will require multiplication of their numerators and denominators separately, 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 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 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
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, bear in mind that you have to follow the exponential rule, the distributive property, and PEMDAS rules in addition to the concept of multiplication of algebraic expressions. Ultimately, make sure that every term on your expression is in its lowest form.
What is the difference between solving an equation and simplifying an expression?
Simplifying and solving equations are quite different, but, they can be part of the same process the same process because you have to simplify expressions before solving them.
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