Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions are one of the most challenging for beginner pupils in their first years of college or even in high school.
Still, learning how to deal with these equations is essential because it is primary knowledge that will help them eventually be able to solve higher math and complicated problems across various industries.
This article will share everything you need to know simplifying expressions. We’ll cover the proponents of simplifying expressions and then verify our skills via some practice questions.
How Do You Simplify Expressions?
Before learning how to simplify expressions, you must understand what expressions are at their core.
In arithmetics, expressions are descriptions that have at least two terms. These terms can contain variables, numbers, or both and can be linked through addition or subtraction.
For example, let’s take a look at 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 include variables, coefficients, and sometimes constants, are also known as polynomials.
Simplifying expressions is essential because it opens up the possibility of learning how to solve them. Expressions can be expressed in convoluted ways, and without simplifying them, everyone will have a difficult time trying to solve them, with more chance for error.
Of course, all expressions will vary in 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, 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 states that the order of operations for expressions.
Parentheses. Solve equations between the parentheses first by applying addition or applying subtraction. If there are terms right outside the parentheses, use the distributive property to apply multiplication the term outside with the one inside.
Exponents. Where workable, use the exponent properties to simplify the terms that include exponents.
Multiplication and Division. If the equation calls for it, use the multiplication and division principles to simplify like terms that are applicable.
Addition and subtraction. Finally, add or subtract the simplified terms of the equation.
Rewrite. Ensure that there are no more like terms to simplify, and then rewrite the simplified equation.
The Rules For Simplifying Algebraic Expressions
Along with the PEMDAS rule, there are a few additional principles you must 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 leave 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 keeping the x as it is.
Parentheses containing another expression outside of them need to use the distributive property. The distributive property allows you to simplify terms on the 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 referred to as the property of multiplication. When two separate expressions within parentheses are multiplied, the distribution rule is applied, and all unique term will have to be multiplied by the other terms, resulting in 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 outside 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. Like in this example: -(8x + 2) will turn into -8x - 2.
Likewise, a plus sign outside the parentheses denotes that it will have distribution applied to the terms inside. However, this means that you are able to remove the parentheses and write the expression as is due to the fact that the plus sign doesn’t alter anything when distributed.
How to Simplify Expressions with Exponents
The prior rules were simple enough to follow as they only applied to rules that impact simple terms with numbers and variables. Still, there are additional rules that you need to follow when dealing with expressions with exponents.
Next, we will review the properties of exponents. 8 principles impact how we process exponentials, which are the following:
Zero Exponent Rule. This principle states that any term with the exponent of 0 equals 1. Or a0 = 1.
Identity Exponent Rule. Any term with a 1 exponent will not alter the value. Or a1 = a.
Product Rule. When two terms with the same variables are multiplied, their product will add their two exponents. This is expressed in the formula am × an = am+n
Quotient Rule. When two terms with the same variables are divided by each other, their quotient applies subtraction to their respective exponents. This is written as 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 that already has an exponent, the term will end up 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 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 denominator and numerator will take the exponent given, (a/b)m = am/bm.
Simplifying Expressions with the Distributive Property
The distributive property is the property 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 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 result 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 keep in mind.
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 states 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 should be expressed in the expression. Refer to the PEMDAS property and make sure that no two terms contain the same variables.
These are the exact principles that you can apply when simplifying any real numbers, whether they are square roots, binomials, decimals, quadratic equations, logarithms, or linear equations.
Practice Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
Here, the properties that should be noted first are PEMDAS and the distributive property. The distributive property will distribute 4 to the expressions on the inside of the parentheses, while PEMDAS will govern the order of simplification.
Due to 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 all the terms with the same variables, and all term should be in its most simplified form.
28x + 28 - 3y
Rearrange the equation like this:
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 within parentheses, and in this scenario, that expression also needs the distributive property. In this scenario, the term y/4 will need to be distributed within the two terms within the parentheses, as seen here.
1/3x + y/4(5x) + y/4(2)
Here, let’s set aside the first term for now and simplify the terms with factors assigned to them. Since we know from PEMDAS that fractions will need to multiply 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 utilized 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 tells us 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
Since there are no more 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 are required to follow the exponential rule, the distributive property, and PEMDAS rules and the concept of multiplication of algebraic expressions. Ultimately, make sure that every term on your expression is in its most simplified form.
How does solving equations differ from simplifying expressions?
Simplifying and solving equations are quite different, but, they can be incorporated into the same process the same process due to the fact that you must first simplify expressions before you begin solving them.
Let Grade Potential Help You Bone Up On Your Math
Simplifying algebraic equations is one of the most foundational precalculus skills you must study. Getting proficient at simplification strategies and laws will pay rewards when you’re practicing advanced mathematics!
But these concepts and laws can get challenging fast. Grade Potential is here to support you, so there is no need to worry!
Grade Potential Jacksonville offers expert instructors that will get you on top of your skills at your convenience. Our expert teachers will guide you applying mathematical principles in a clear manner to assist.
Connect with us now!
