Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can appear to be challenging for beginner students in their primary years of high school or college.
However, understanding how to deal with these equations is essential because it is primary knowledge that will help them move on to higher arithmetics and complex problems across various industries.
This article will share everything you should review to know simplifying expressions. We’ll review the laws of simplifying expressions and then validate what we've learned through some practice problems.
How Does Simplifying Expressions Work?
Before learning how to simplify them, you must understand what expressions are at their core.
In arithmetics, expressions are descriptions that have at least two terms. These terms can combine variables, numbers, or both and can be linked through subtraction or addition.
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 terms consist of both numbers (8 and 2) and variables (x and y).
Expressions consisting of coefficients, variables, and occasionally constants, are also called polynomials.
Simplifying expressions is essential because it opens up the possibility of learning how to solve them. Expressions can be expressed in intricate ways, and without simplification, anyone will have a tough time attempting to solve them, with more possibility for a mistake.
Obviously, each expression be different in how they are simplified depending on what terms they incorporate, but there are common steps that can be applied to all rational expressions of real numbers, regardless of whether they are square roots, logarithms, or otherwise.
These steps are known as the PEMDAS rule, or parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule shows us the order of operations for expressions.
Parentheses. Simplify equations between the parentheses first by adding or applying subtraction. If there are terms right outside the parentheses, use the distributive property to multiply the term outside with the one inside.
Exponents. Where possible, use the exponent principles 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 are applicable.
Addition and subtraction. Then, add or subtract the resulting terms of the equation.
Rewrite. Ensure that there are no remaining like terms that need to be simplified, then rewrite the simplified equation.
The Rules For Simplifying Algebraic Expressions
Beyond the PEMDAS principle, there are a few more properties you should be aware 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 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 keeping the variable x as it is.
Parentheses containing another expression outside of them need to utilize the distributive property. The distributive property prompts you to simplify terms 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 known as the principle of multiplication. When two separate expressions within parentheses are multiplied, the distribution principle is applied, and each separate term will need to be multiplied by the other terms, resulting in 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 should also need to be distributed, changing the signs of the terms inside the parentheses. For example: -(8x + 2) will turn into -8x - 2.
Similarly, a plus sign right outside the parentheses denotes that it will have distribution applied to the terms inside. But, this means that you can remove the parentheses and write the expression as is because the plus sign doesn’t alter anything when distributed.
How to Simplify Expressions with Exponents
The prior properties were easy enough to implement as they only dealt with principles that impact simple terms with variables and numbers. However, there are a few other rules that you have to apply when working with expressions with exponents.
Next, we will talk about the properties of exponents. 8 properties impact how we process exponents, those are the following:
Zero Exponent Rule. This property states that any term with a 0 exponent equals 1. Or a0 = 1.
Identity Exponent Rule. Any term with a 1 exponent 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 the same variables are divided, their quotient applies subtraction to their two 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 already with 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 possess unique 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 says that any term multiplied by an expression within parentheses must be multiplied by all of the expressions inside. Let’s watch 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 resulting expression is 6x + 10.
Simplifying Expressions with Fractions
Certain expressions can consist of fractions, and just as with exponents, expressions with fractions also have multiple rules that you have to follow.
When an expression has fractions, here is what to remember.
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 tells 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 state should be expressed in the expression. Use the PEMDAS property and ensure that no two terms share matching variables.
These are the same properties that you can apply when simplifying any real numbers, whether they are decimals, square roots, binomials, logarithms, linear equations, or quadratic equations.
Practice Examples for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this example, the principles that should 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 decide on 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 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 expresses that the first in order should be expressions within parentheses, and in this case, that expression also needs the distributive property. In this scenario, the term y/4 will need to be distributed to the two terms on the inside of 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 attached to them. Since 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 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 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 simplify, this becomes our final answer.
Simplifying Expressions FAQs
What should I bear in mind when simplifying expressions?
When simplifying algebraic expressions, remember that you are required to follow the exponential rule, the distributive property, and PEMDAS rules and the concept of multiplication of algebraic expressions. In the end, make sure that every term on your expression is in its lowest form.
How are simplifying expressions and solving equations different?
Solving equations and simplifying expressions are very different, although, they can be incorporated into the same process the same process because you first need to simplify expressions before you solve them.
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