Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be intimidating for new pupils in their primary years of college or even in high school.
However, learning how to process these equations is important because it is primary knowledge that will help them move on to higher arithmetics and complex problems across multiple industries.
This article will discuss everything you must have to master simplifying expressions. We’ll cover the proponents of simplifying expressions and then validate our comprehension via some sample questions.
How Do You Simplify Expressions?
Before you can learn how to simplify expressions, you must grasp what expressions are at their core.
In arithmetics, expressions are descriptions that have no less than two terms. These terms can include variables, numbers, or both and can be connected through subtraction or addition.
To give an example, let’s take a look at the following expression.
8x + 2y - 3
This expression contains 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 known as polynomials.
Simplifying expressions is important because it lays the groundwork for learning how to solve them. Expressions can be expressed in convoluted ways, and without simplifying them, you will have a tough time attempting to solve them, with more opportunity for solving them incorrectly.
Of course, each expression be different regarding how they are simplified based on what terms they incorporate, but there are general 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 called the PEMDAS rule, an abbreviation for parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule shows us the order of operations for expressions.
Parentheses. Simplify equations within the parentheses first by using addition or subtracting. If there are terms just outside the parentheses, use the distributive property to apply multiplication the term outside with the one inside.
Exponents. Where workable, use the exponent rules to simplify the terms that have exponents.
Multiplication and Division. If the equation requires it, utilize multiplication and division to simplify like terms that apply.
Addition and subtraction. Lastly, add or subtract the remaining 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 sequence, there are a few more rules you must be informed of when simplifying 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 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 on the 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 known as the principle of multiplication. When two stand-alone expressions within parentheses are multiplied, the distributive principle is applied, and all unique term will will require multiplication by the other terms, resulting in each set of equations, common factors of each other. 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 will also need to have distribution applied, changing the signs of the terms on the inside of the parentheses. For example: -(8x + 2) will turn into -8x - 2.
Likewise, a plus sign right outside the parentheses will mean 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 due to the fact that the plus sign doesn’t change anything when distributed.
How to Simplify Expressions with Exponents
The prior principles were straight-forward enough to implement as they only applied to rules that affect simple terms with variables and numbers. Despite that, there are additional rules that you need to follow when dealing with expressions with exponents.
In this section, we will talk about the properties of exponents. Eight properties influence how we utilize exponentials, that includes the following:
Zero Exponent Rule. This property 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 doesn'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 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 applicable 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 already with an exponent, the term will end up being the 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 different variables should 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 property that shows us that any term multiplied by an expression within parentheses needs be multiplied by all of the expressions on the inside. Let’s see 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 expression then becomes 6x + 10.
Simplifying Expressions with Fractions
Certain expressions contain fractions, and just like with exponents, expressions with fractions also have some rules that you have to follow.
When an expression includes 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 shows us 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 should be expressed in the expression. Apply the PEMDAS rule and be sure that no two terms have the same variables.
These are the same principles that you can apply when simplifying any real numbers, whether they are binomials, decimals, square roots, logarithms, linear equations, or quadratic equations.
Sample Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this example, the rules that need to be noted first are PEMDAS and the distributive property. The distributive property will distribute 4 to all other expressions inside of the parentheses, while PEMDAS will decide on the order of simplification.
As a result of the distributive property, the term outside 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 all term should be in its lowest form.
28x + 28 - 3y
Rearrange the equation as follows:
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 requires the distributive property. In this example, the term y/4 must be distributed within 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 the moment and simplify the terms with factors assigned to them. Because we know from PEMDAS that fractions will need to multiply their denominators and numerators separately, we will then have:
y/4 * 5x/1
The expression 5x/1 is used to keep things simple 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 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
Due to the fact that 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 have to follow the exponential rule, the distributive property, and PEMDAS rules as well as the principle of multiplication of algebraic expressions. Finally, ensure 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 vastly different, but, they can be part of the same process the same process since you have to simplify expressions before you begin solving them.
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