Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions are one of the most challenging for budding pupils in their early years of college or even in high school.
Nevertheless, understanding how to deal with these equations is essential because it is foundational information that will help them navigate higher mathematics and complex problems across multiple industries.
This article will go over everything you need to know simplifying expressions. We’ll cover the proponents of simplifying expressions and then verify what we've learned via some sample problems.
How Do You Simplify Expressions?
Before learning how to simplify them, you must grasp 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 linked through addition or subtraction.
As an example, let’s review the following expression.
8x + 2y - 3
This expression combines three terms; 8x, 2y, and 3. The first two terms include both numbers (8 and 2) and variables (x and y).
Expressions that include variables, coefficients, and sometimes constants, are also referred to as polynomials.
Simplifying expressions is important because it lays the groundwork for understanding how to solve them. Expressions can be expressed in convoluted ways, and without simplification, everyone will have a hard time trying to solve them, with more opportunity for error.
Of course, every expression vary regarding how they are simplified depending on what terms they contain, but there are general steps that apply to all rational expressions of real numbers, whether they are logarithms, square roots, etc.
These steps are refered to as the PEMDAS rule, short for parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule states that the order of operations for expressions.
Parentheses. Simplify equations between the parentheses first by applying addition or subtracting. If there are terms just outside the parentheses, use the distributive property to apply multiplication the term on the outside with the one inside.
Exponents. Where feasible, use the exponent rules to simplify the terms that include exponents.
Multiplication and Division. If the equation calls for it, use multiplication and division to simplify like terms that apply.
Addition and subtraction. Finally, add or subtract the simplified terms in the equation.
Rewrite. Ensure that there are no additional like terms that require simplification, and then rewrite the simplified equation.
Here are the Requirements For Simplifying Algebraic Expressions
Beyond 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 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 variable x as it is.
Parentheses containing another expression outside of them need to use the distributive property. The distributive property allows 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 referred to as the property of multiplication. When two separate expressions within parentheses are multiplied, the distributive rule kicks in, and every separate term will will require multiplication by the other terms, resulting in each set of equations, common factors of one another. For example: (a + b)(c + d) = a(c + d) + b(c + d).
A negative sign right outside of an expression in parentheses denotes that the negative expression should also need to have distribution applied, changing the signs of the terms inside the parentheses. As is the case in this example: -(8x + 2) will turn into -8x - 2.
Likewise, a plus sign on the outside of the parentheses means that it will be distributed to the terms on the inside. However, this means that you are able to remove the parentheses and write the expression as is owing to the fact that the plus sign doesn’t alter anything when distributed.
How to Simplify Expressions with Exponents
The prior rules were straight-forward enough to follow as they only applied to rules that impact simple terms with variables and numbers. However, there are more rules that you need to apply when dealing with expressions with exponents.
In this section, we will review the principles of exponents. 8 properties affect how we deal with exponents, those are the following:
Zero Exponent Rule. This property states that any term with the exponent of 0 equals 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 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 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 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 principle that says that any term multiplied by an expression on the inside of a parentheses must be multiplied by all of the expressions within. Let’s watch 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 can consist of fractions, and just like with exponents, expressions with fractions also have multiple rules that you must follow.
When an expression contains 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 usually be the power of the quotient rule, which will subtract the exponents of the numerators and denominators.
Simplification. Only fractions at their lowest form should be included in the expression. Refer to the PEMDAS property and be sure that no two terms possess matching variables.
These are the exact rules 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.
Here, the rules that should be noted first are PEMDAS and the distributive property. The distributive property will distribute 4 to the expressions on the inside of 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 the individual terms inside.
4(2x) + 4(5x) + 4(7) - 3y
8x + 20x + 28 - 3y
When simplifying equations, be sure to add all the terms with matching variables, and each term should be in its lowest 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 you should begin with expressions on the inside of parentheses, and in this example, that expression also needs the distributive property. Here, the term y/4 should be distributed to the two terms within the parentheses, as seen here.
1/3x + y/4(5x) + y/4(2)
Here, let’s set aside the first term for the moment and simplify the terms with factors assigned to them. Remember 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 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 used 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, 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 other like terms to be simplified, this becomes our final answer.
Simplifying Expressions FAQs
What should I bear in mind when simplifying expressions?
When simplifying algebraic expressions, keep in mind that you must follow PEMDAS, the exponential rule, and the distributive property rules in addition to the principle of multiplication of algebraic expressions. Ultimately, 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 very different, but, they can be combined the same process because you have to simplify expressions before you begin solving them.
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