Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be intimidating for budding students in their first years of high school or college.
However, learning how to process these equations is critical because it is primary information that will help them navigate higher arithmetics and advanced problems across different industries.
This article will share everything you need to know simplifying expressions. We’ll learn the principles of simplifying expressions and then validate our skills with some practice problems.
How Do You Simplify Expressions?
Before you can be taught how to simplify expressions, you must learn what expressions are to begin with.
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 addition or subtraction.
As 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 consist of 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 lays the groundwork for learning how to solve them. Expressions can be expressed in complicated ways, and without simplifying them, you will have a difficult time attempting to solve them, with more possibility for error.
Undoubtedly, each expression vary regarding how they're simplified based on what terms they incorporate, but there are common steps that are applicable to all rational expressions of real numbers, whether they are square roots, logarithms, or otherwise.
These steps are known as the PEMDAS rule, short for parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule declares the order of operations for expressions.
Parentheses. Simplify equations inside the parentheses first by using addition or subtracting. If there are terms right outside the parentheses, use the distributive property to multiply the term outside with the one on the inside.
Exponents. Where workable, use the exponent rules to simplify the terms that contain exponents.
Multiplication and Division. If the equation calls for it, use the multiplication and division principles to simplify like terms that apply.
Addition and subtraction. Then, add or subtract the resulting terms in the equation.
Rewrite. Ensure that there are no additional like terms to simplify, then rewrite the simplified equation.
The Requirements For Simplifying Algebraic Expressions
In addition to the PEMDAS principle, there are a few additional principles you must be informed of when simplifying algebraic expressions.
You can only apply simplification to 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 equation 8x + 2x can be simplified to 10x by applying addition to the coefficients 8 and 2 and retaining the variable x as it is.
Parentheses that contain another expression directly outside of them need to apply the distributive property. The distributive property allows you to simplify terms outside of parentheses by distributing them to the terms inside, for example: a(b+c) = ab + ac.
An extension of the distributive property is referred to as the concept of multiplication. When two separate expressions within parentheses are multiplied, the distributive rule applies, and all separate term will have to be multiplied 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 right outside of an expression in parentheses indicates that the negative expression must 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 means that it will be distributed to the terms on the inside. Despite that, this means that you are able to 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 previous properties were simple enough to use as they only applied to rules that affect simple terms with numbers and variables. Despite that, there are more rules that you have to follow when dealing with expressions with exponents.
Here, we will review the properties of exponents. Eight properties impact how we utilize exponentials, which are the following:
Zero Exponent Rule. This property states that any term with a 0 exponent 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, 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 subtracts their two 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 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 that were applied to it, or (am)n = amn.
Power of a Product Rule. An exponent applied to two terms that have unique variables should be applied to the required variables, or (ab)m = am * bm.
Power of a Quotient Rule. In fractional exponents, both the numerator and denominator 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 denotes that any term multiplied by an expression on the inside of a parentheses needs be multiplied by all of the expressions within. Let’s see 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 like with exponents, expressions with fractions also have some rules that you have to follow.
When an expression contains fractions, here is what to remember.
Distributive property. The distributive property a(b+c) = ab + ac, when applied to fractions, will multiply fractions one at a time by their numerators and denominators.
Laws of exponents. This shows us that fractions will more likely 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. Refer to the PEMDAS rule and ensure that no two terms have the same variables.
These are the exact properties that you can apply when simplifying any real numbers, whether they are binomials, decimals, square roots, 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 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, remember to add all the terms with matching 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 example, that expression also requires the distributive property. In this example, the term y/4 must be distributed to the two terms inside the parentheses, as seen here.
1/3x + y/4(5x) + y/4(2)
Here, let’s put aside the first term for now and simplify the terms with factors associated with them. Because we know from PEMDAS that fractions will need to multiply 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 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, which tells us 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
Since 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, remember that you are required to follow the distributive property, PEMDAS, and the exponential rule rules and the rule 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?
Simplifying and solving equations are quite different, although, they can be part of the same process the same process because you have to simplify expressions before you solve them.
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