Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be challenging for budding students in their first years of college or even in high school.
Still, understanding how to handle these equations is essential because it is primary knowledge that will help them eventually be able to solve higher math and advanced problems across multiple industries.
This article will share everything you should review to know simplifying expressions. We’ll review the laws of simplifying expressions and then verify our comprehension with some practice questions.
How Do I Simplify an Expression?
Before you can be taught how to simplify expressions, you must understand what expressions are to begin with.
In mathematics, expressions are descriptions that have at least two terms. These terms can include variables, numbers, or both and can be connected through subtraction or addition.
For example, let’s go over the following expression.
8x + 2y - 3
This expression includes three terms; 8x, 2y, and 3. The first two terms contain both numbers (8 and 2) and variables (x and y).
Expressions containing coefficients, variables, and occasionally constants, are also called polynomials.
Simplifying expressions is important because it opens up the possibility of grasping how to solve them. Expressions can be written in intricate ways, and without simplification, everyone will have a tough time attempting to solve them, with more chance for solving them incorrectly.
Obviously, every expression vary regarding how they are simplified based on what terms they include, but there are typical steps that apply to all rational expressions of real numbers, 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. Resolve equations within the parentheses first by adding or using subtraction. If there are terms just outside the parentheses, use the distributive property to apply multiplication the term 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 calls for it, utilize multiplication or division rules to simplify like terms that are applicable.
Addition and subtraction. Lastly, add or subtract the simplified terms in the equation.
Rewrite. Make sure that there are no remaining like terms to simplify, and rewrite the simplified equation.
Here are the Properties For Simplifying Algebraic Expressions
Along with the PEMDAS principle, there are a few more rules you must be aware of when simplifying algebraic expressions.
You can only simplify terms with common variables. When applying addition to these terms, add the coefficient numbers and keep 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 keeping the 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 on the outside of parentheses by distributing them to the terms on the inside, as shown here: a(b+c) = ab + ac.
An extension of the distributive property is called the property of multiplication. When two separate expressions within parentheses are multiplied, the distribution rule is applied, and each unique term will will require multiplication 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 directly outside of an expression in parentheses means that the negative expression must 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.
Likewise, a plus sign right outside the parentheses means that it will be distributed to the terms inside. However, this means that you can eliminate 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 previous principles were straight-forward enough to use as they only dealt with rules that affect simple terms with variables and numbers. Still, there are additional rules that you need to implement when dealing with expressions with exponents.
In this section, we will talk about the principles of exponents. 8 rules affect how we deal with exponents, those are the following:
Zero Exponent Rule. This rule states that any term with a 0 exponent is equivalent to 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 multiplied, 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 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 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 applied to it, or (am)n = amn.
Power of a Product Rule. An exponent applied to two terms that have differing 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 acquire the exponent given, (a/b)m = am/bm.
Simplifying Expressions with the Distributive Property
The distributive property is the principle that shows us that any term multiplied by an expression within parentheses needs be multiplied by all of the expressions inside. 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 expression then becomes 6x + 10.
How to Simplify Expressions with Fractions
Certain expressions contain fractions, and just as with exponents, expressions with fractions also have some rules that you have 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 separately by their denominators and numerators.
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 state should be written in the expression. Use the PEMDAS property and ensure that no two terms contain matching variables.
These are the same rules that you can apply when simplifying any real numbers, whether they are square roots, binomials, decimals, logarithms, linear equations, or quadratic equations.
Sample Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
Here, the principles that must 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 dictate the order of simplification.
Due to the distributive property, the term on the 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 all the terms with the same variables, and each 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 expresses that the the order should start with expressions within parentheses, and in this case, that expression also necessitates the distributive property. In this scenario, the term y/4 must be distributed within the two terms inside the parentheses, as seen in this example.
1/3x + y/4(5x) + y/4(2)
Here, let’s put aside the first term for now and simplify the terms with factors assigned to them. Remember we know from PEMDAS that fractions will need to multiply their denominators and numerators individually, we will then have:
y/4 * 5x/1
The expression 5x/1 is used to keep things simple 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 utilized to distribute all terms 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 the same variables multiplied together and multiply their coefficients. This gives us:
24x3 + 4x2 + 18xy + 3y
Because there are no other like terms to be simplified, this becomes our final answer.
Simplifying Expressions FAQs
What should I keep in mind when simplifying expressions?
When simplifying algebraic expressions, remember that you have to follow the exponential rule, the distributive property, and PEMDAS rules in addition to the concept of multiplication of algebraic expressions. In the end, 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 very different, but, they can be combined the same process since you first need to simplify expressions before you solve them.
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