Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can appear to be intimidating for budding pupils in their early years of college or even in high school.
However, grasping how to process these equations is important because it is primary information that will help them move on to higher arithmetics and complicated problems across different industries.
This article will go over everything you need to master simplifying expressions. We’ll learn the laws of simplifying expressions and then verify what we've learned via some practice questions.
How Do I Simplify an Expression?
Before you can be taught how to simplify expressions, you must understand what expressions are at their core.
In mathematics, expressions are descriptions that have a minimum of two terms. These terms can include numbers, variables, or both and can be linked through addition or subtraction.
As an 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 incorporate variables, coefficients, and sometimes constants, are also referred to as polynomials.
Simplifying expressions is essential because it opens up the possibility of understanding how to solve them. Expressions can be expressed in convoluted ways, and without simplification, anyone will have a tough time attempting to solve them, with more opportunity for a mistake.
Obviously, every expression be different in how they are simplified based on what terms they incorporate, but there are general steps that apply to all rational expressions of real numbers, regardless of whether they are square roots, logarithms, or otherwise.
These steps are refered to as the PEMDAS rule, or parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule states that the order of operations for expressions.
Parentheses. Solve equations within the parentheses first by using 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 inside.
Exponents. Where workable, use the exponent properties to simplify the terms that have exponents.
Multiplication and Division. If the equation necessitates it, utilize multiplication or division rules to simplify like terms that apply.
Addition and subtraction. Then, use addition or subtraction the remaining terms in the equation.
Rewrite. Ensure that there are no more like terms that need to be simplified, and then rewrite the simplified equation.
Here are the Properties For Simplifying Algebraic Expressions
Beyond the PEMDAS rule, there are a few additional rules you should be informed 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 applying addition to the coefficients 8 and 2 and keeping the x as it is.
Parentheses that contain another expression 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, as shown here: a(b+c) = ab + ac.
An extension of the distributive property is called the principle of multiplication. When two separate expressions within parentheses are multiplied, the distributive principle kicks in, and every individual term will have to be multiplied by the other terms, resulting in each set of equations, common factors of one another. For example: (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 must also need to have distribution applied, changing the signs of the terms on the inside of the parentheses. As is the case in this example: -(8x + 2) will turn into -8x - 2.
Likewise, a plus sign right outside the parentheses denotes that it will be distributed to the terms on the inside. But, this means that you should eliminate the parentheses and write the expression as is because the plus sign doesn’t change anything when distributed.
How to Simplify Expressions with Exponents
The previous properties were simple enough to follow as they only dealt with properties that impact simple terms with numbers and variables. Still, there are additional rules that you need to apply when working with expressions with exponents.
Here, we will talk about 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 is equal to 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 apply multiplication, 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, their quotient will subtract 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 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 have 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 take the exponent given, (a/b)m = am/bm.
Simplifying Expressions with the Distributive Property
The distributive property is the principle that states that any term multiplied by an expression within parentheses needs be multiplied by all of the expressions inside. Let’s witness 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 expression then becomes 6x + 10.
Simplifying Expressions with Fractions
Certain expressions can consist of fractions, and just as with exponents, expressions with fractions also have some rules that you must 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 separately by their numerators and denominators.
Laws of exponents. This shows us that fractions will typically be the power of the quotient rule, which will apply subtraction to the exponents of the numerators and denominators.
Simplification. Only fractions at their lowest should be expressed in the expression. Use the PEMDAS property and be sure that no two terms possess the same variables.
These are the exact properties that you can apply when simplifying any real numbers, whether they are decimals, square roots, binomials, linear equations, quadratic equations, and even logarithms.
Sample Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
Here, the principles that need to be noted first are PEMDAS and the distributive property. The distributive property will distribute 4 to all the expressions inside of the parentheses, while PEMDAS will govern the order of simplification.
Due to the distributive property, the term 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, you should add the terms with matching variables, and all term should be in its lowest 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 states that the the order should start with expressions inside parentheses, and in this scenario, that expression also necessitates the distributive property. In this scenario, the term y/4 must 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 attached to them. Since we know from PEMDAS that fractions will require multiplication of their numerators and denominators individually, we will then have:
y/4 * 5x/1
The expression 5x/1 is used for simplicity because 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 each 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 similar variables multiplied together and multiply their coefficients. This gives us:
24x3 + 4x2 + 18xy + 3y
Since there are no more 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 have to follow the exponential rule, the distributive property, and PEMDAS rules as well as the rule of multiplication of algebraic expressions. Ultimately, ensure that every term on your expression is in its most simplified form.
What is the difference between solving an equation and simplifying an expression?
Simplifying and solving equations are quite different, although, they can be combined the same process since you must first simplify expressions before you solve them.
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