Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be intimidating for new students in their early years of college or even in high school.
However, understanding how to deal with these equations is important because it is primary knowledge that will help them eventually be able to solve higher mathematics and advanced problems across different industries.
This article will go over everything you need to know simplifying expressions. We’ll learn the principles of simplifying expressions and then verify our comprehension via some sample problems.
How Do I Simplify an Expression?
Before learning how to simplify expressions, you must understand what expressions are to begin with.
In arithmetics, expressions are descriptions that have a minimum of two terms. These terms can include numbers, variables, or both and can be linked through subtraction or addition.
For example, let’s review the following expression.
8x + 2y - 3
This expression includes 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 essential because it opens up the possibility of learning how to solve them. Expressions can be expressed in complicated ways, and without simplification, you will have a difficult time attempting to solve them, with more opportunity for solving them incorrectly.
Undoubtedly, each expression be different regarding how they're simplified based on what terms they include, but there are typical steps that can be applied to all rational expressions of real numbers, whether they are square roots, logarithms, or otherwise.
These steps are refered to as 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 inside the parentheses first by adding or using subtraction. If there are terms right outside the parentheses, use the distributive property to multiply the term on the outside with the one inside.
Exponents. Where feasible, use the exponent properties to simplify the terms that have exponents.
Multiplication and Division. If the equation calls for it, utilize multiplication and division to simplify like terms that apply.
Addition and subtraction. Lastly, add or subtract the simplified terms of the equation.
Rewrite. Make sure that there are no more like terms to simplify, and then rewrite the simplified equation.
The Properties For Simplifying Algebraic Expressions
In addition to the PEMDAS sequence, there are a few more principles you should be aware of when dealing 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 applying addition to the coefficients 8 and 2 and leaving the x as it is.
Parentheses containing another expression outside of them need to apply the distributive property. The distributive property prompts 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 called the principle of multiplication. When two separate expressions within parentheses are multiplied, the distributive property is applied, and each unique 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 outside an expression in parentheses indicates that the negative expression will also need to have distribution applied, changing the signs of the terms on the inside of the parentheses. Like in this example: -(8x + 2) will turn into -8x - 2.
Similarly, a plus sign on the outside of the parentheses denotes that it will have distribution applied to the terms on the inside. However, this means that you should remove 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 prior rules were easy enough to implement as they only dealt with properties that affect simple terms with variables and numbers. Despite that, there are additional rules that you have to follow when dealing with exponents and expressions.
Next, we will talk about the properties of exponents. 8 properties influence how we process exponentials, which are the following:
Zero Exponent Rule. This property states that any term with the exponent of 0 is equivalent 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 equivalent variables are multiplied by each other, their product will add their two exponents. This is expressed in the formula am × an = am+n
Quotient Rule. When two terms with matching variables are divided, their quotient subtracts their respective exponents. This is expressed in 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 expressed with 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 end up 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 possess differing variables needs to 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.
Simplifying Expressions with the Distributive Property
The distributive property is the property that shows us that any term multiplied by an expression within parentheses needs be multiplied by all of the expressions within. Let’s witness 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 multiple rules that you must follow.
When an expression contains fractions, here is 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 states that fractions will typically 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 state should be included in the expression. Refer to the PEMDAS principle and be sure that no two terms contain matching variables.
These are the same rules that you can apply when simplifying any real numbers, whether they are binomials, decimals, square roots, quadratic equations, logarithms, or linear equations.
Practice Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this example, the rules that should be noted first are the PEMDAS and the distributive property. The distributive property will distribute 4 to all other expressions on the inside of the parentheses, while PEMDAS will govern the order of simplification.
Because of 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, you should add all the terms with matching variables, and each 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 states that the you should begin with expressions within parentheses, and in this example, that expression also needs the distributive property. Here, the term y/4 will need to be distributed amongst the two terms inside the parentheses, as follows.
1/3x + y/4(5x) + y/4(2)
Here, let’s put aside the first term for the moment and simplify the terms with factors assigned to them. Since 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 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 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, meaning 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
Because there are no more like terms to simplify, this becomes our final answer.
Simplifying Expressions FAQs
What should I remember when simplifying expressions?
When simplifying algebraic expressions, bear in mind that you are required to obey PEMDAS, the exponential rule, and the distributive property 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.
How does solving equations differ from simplifying expressions?
Solving equations and simplifying expressions are very different, but, they can be incorporated into the same process the same process since you must first simplify expressions before you solve them.
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