Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions are one of the most challenging for budding students in their first years of high school or college.
Still, grasping how to handle these equations is essential because it is primary knowledge that will help them move on to higher mathematics and advanced problems across multiple industries.
This article will share everything you need to master simplifying expressions. We’ll review the proponents of simplifying expressions and then validate our skills with some practice questions.
How Do You Simplify Expressions?
Before learning how to simplify them, you must understand what expressions are in the first place.
In arithmetics, expressions are descriptions that have a minimum of two terms. These terms can contain numbers, variables, or both and can be connected through subtraction or addition.
To give an example, let’s go over the following expression.
8x + 2y - 3
This expression combines three terms; 8x, 2y, and 3. The first two terms contain both numbers (8 and 2) and variables (x and y).
Expressions that incorporate variables, coefficients, and sometimes constants, are also called polynomials.
Simplifying expressions is essential because it lays the groundwork for grasping how to solve them. Expressions can be expressed in intricate ways, and without simplification, you will have a hard time trying to solve them, with more possibility for error.
Obviously, each expression vary regarding how they are simplified based on what terms they incorporate, but there are typical steps that apply 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 declares the order of operations for expressions.
Parentheses. Resolve equations between the parentheses first by adding or subtracting. 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 include exponents.
Multiplication and Division. If the equation necessitates it, use the multiplication and division principles to simplify like terms that are applicable.
Addition and subtraction. Then, add or subtract the remaining terms in the equation.
Rewrite. Make sure that there are no additional like terms that need to be simplified, then rewrite the simplified equation.
Here are the Properties For Simplifying Algebraic Expressions
In addition to the PEMDAS sequence, there are a few more principles you need to 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 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 leaving the x as it is.
Parentheses that include another expression directly 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 on the 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 rule applies, and every unique term will need to be multiplied by the other terms, making each set of equations, common factors of one another. For example: (a + b)(c + d) = a(c + d) + b(c + d).
A negative sign directly outside of an expression in parentheses indicates that the negative expression should also need to be distributed, 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 right outside the parentheses means that it will be distributed to the terms on the inside. However, this means that you should remove the parentheses and write the expression as is due to the fact that the plus sign doesn’t change anything when distributed.
How to Simplify Expressions with Exponents
The prior rules were straight-forward enough to implement as they only dealt with properties that impact simple terms with variables and numbers. Despite that, there are a few other rules that you need to implement when dealing with exponents and expressions.
Here, we will review the laws of exponents. Eight principles impact how we utilize exponentials, those are the following:
Zero Exponent Rule. This principle 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 the same variables are multiplied by each other, their product will add their exponents. This is written as am × an = am+n
Quotient Rule. When two terms with the same variables are divided, their quotient applies subtraction to their respective exponents. This is expressed in the formula am/an = am-n.
Negative Exponents Rule. Any term with a negative exponent equals 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 that already has an exponent, the term will end up being the 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 needs to be applied to the required 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.
How to Simplify Expressions with the Distributive Property
The distributive property is the rule that states that any term multiplied by an expression on the inside of a parentheses must be multiplied by all of the expressions within. 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 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 has 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 denominators and numerators.
Laws of exponents. This shows us that fractions will more likely be the power of the quotient rule, which will subtract the exponents of the denominators and numerators.
Simplification. Only fractions at their lowest state should be written in the expression. Use the PEMDAS property and be sure that no two terms have the same variables.
These are the same properties that you can apply when simplifying any real numbers, whether they are decimals, square roots, binomials, quadratic equations, logarithms, or linear equations.
Practice Examples for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this case, the principles that should 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 decide on the order of simplification.
As a result of the distributive property, the term 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 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 states that the first in order should be expressions inside parentheses, and in this scenario, that expression also needs the distributive property. Here, the term y/4 should be distributed to 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 the moment 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 used to distribute every term 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, meaning that we’ll have to add the exponents of two exponential expressions with like 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 keep in mind when simplifying expressions?
When simplifying algebraic expressions, keep in mind that you must follow the exponential rule, the distributive property, and PEMDAS rules as well as the rule of multiplication of algebraic expressions. Finally, 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?
Solving and simplifying expressions are very different, although, they can be incorporated into the same process the same process because you must first simplify expressions before you solve them.
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