Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be scary for new pupils in their first years of college or even in high school.
Still, understanding how to process these equations is critical because it is foundational knowledge that will help them eventually be able to solve higher arithmetics and complicated problems across various industries.
This article will discuss everything you should review to learn simplifying expressions. We’ll cover the laws of simplifying expressions and then validate our comprehension through some practice questions.
How Does Simplifying Expressions Work?
Before you can learn how to simplify them, you must learn what expressions are in the first place.
In mathematics, expressions are descriptions that have a minimum of two terms. These terms can include variables, numbers, or both and can be connected through addition or subtraction.
To give an example, let’s go over the following expression.
8x + 2y - 3
This expression contains 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 known as polynomials.
Simplifying expressions is essential 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 difficult time trying to solve them, with more chance for solving them incorrectly.
Of course, each expression differ regarding how they're simplified depending on what terms they contain, but there are common steps that can be applied to all rational expressions of real numbers, regardless of 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 applying addition or using subtraction. If there are terms just outside the parentheses, use the distributive property to apply multiplication the term on the outside with the one on the inside.
Exponents. Where feasible, use the exponent properties 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. Lastly, use addition or subtraction the resulting terms of the equation.
Rewrite. Make sure that there are no more like terms to simplify, and rewrite the simplified equation.
The Properties For Simplifying Algebraic Expressions
Along with the PEMDAS rule, 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 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 adding coefficients 8 and 2 and retaining the variable x as it is.
Parentheses that include another expression outside of them need to utilize the distributive property. The distributive property gives you the ability to to simplify terms on the outside of parentheses by distributing them to the terms on the inside, or as follows: a(b+c) = ab + ac.
An extension of the distributive property is referred to as the principle of multiplication. When two stand-alone expressions within parentheses are multiplied, the distributive rule kicks in, and every separate term will need to be multiplied by the other terms, resulting in each set of equations, common factors of each other. For example: (a + b)(c + d) = a(c + d) + b(c + d).
A negative sign outside an expression in parentheses indicates that the negative expression must also need to have distribution applied, changing the signs of the terms inside the parentheses. Like 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 have distribution applied to the terms on the inside. However, this means that you are able to remove 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 properties that impact simple terms with numbers and variables. Still, there are a few other rules that you must apply when dealing with exponents and expressions.
In this section, we will talk about the properties of exponents. Eight rules impact how we utilize exponents, that includes 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 won't change in value. Or a1 = a.
Product Rule. When two terms with equivalent variables are apply multiplication, their product will add their exponents. This is written as am × an = am+n
Quotient Rule. When two terms with the same variables are divided by each other, their quotient subtracts 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 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 applied to it, or (am)n = amn.
Power of a Product Rule. An exponent applied to two terms that have different variables needs to be applied to the appropriate 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 denotes that any term multiplied by an expression within parentheses should be multiplied by all of the expressions within. 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 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 some rules that you need to follow.
When an expression includes fractions, here's 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 numerators and denominators.
Laws of exponents. This shows us that fractions will more likely 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 should be expressed in the expression. Use the PEMDAS property and ensure that no two terms share the same variables.
These are the exact rules that you can apply when simplifying any real numbers, whether they are binomials, decimals, square roots, quadratic equations, logarithms, or linear equations.
Practice Examples for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this example, the principles that must 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.
As a result of the distributive property, the term outside 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 matching variables, and all 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 you should begin with expressions on the inside of parentheses, and in this scenario, that expression also needs the distributive property. In this scenario, the term y/4 will need to be distributed amongst the two terms inside the parentheses, as seen in this example.
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 assigned to them. Because we know from PEMDAS that fractions require multiplication of their numerators and denominators separately, we will then have:
y/4 * 5x/1
The expression 5x/1 is used to keep things simple 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 all terms 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
Due to the fact that there are no remaining like terms to be simplified, 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 follow the exponential rule, the distributive property, and PEMDAS rules in addition to the principle 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?
Solving and simplifying expressions are very different, however, they can be incorporated into the same process the same process because you first need to simplify expressions before you begin solving them.
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