Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can appear to be intimidating for budding students in their primary years of college or even in high school.
However, learning how to deal with these equations is important because it is primary information that will help them eventually be able to solve higher mathematics and complicated problems across multiple industries.
This article will go over everything you should review to know simplifying expressions. We’ll cover the principles of simplifying expressions and then verify what we've learned through some sample questions.
How Do You Simplify Expressions?
Before learning how to simplify them, you must grasp what expressions are to begin with.
In mathematics, expressions are descriptions that have a minimum of two terms. These terms can combine variables, numbers, or both and can be connected through addition or subtraction.
For example, let’s review the following expression.
8x + 2y - 3
This expression includes three terms; 8x, 2y, and 3. The first two terms include both numbers (8 and 2) and variables (x and y).
Expressions consisting of coefficients, variables, and occasionally constants, are also referred to as polynomials.
Simplifying expressions is essential because it opens up the possibility of learning how to solve them. Expressions can be written in convoluted ways, and without simplification, everyone will have a difficult time attempting to solve them, with more opportunity for error.
Obviously, all expressions will be different regarding how they are simplified depending on what terms they incorporate, but there are general steps that apply to all rational expressions of real numbers, regardless of whether they are square roots, logarithms, or otherwise.
These steps are refered to as the PEMDAS rule, an abbreviation for parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule states that the order of operations for expressions.
Parentheses. Resolve equations within 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 on the inside.
Exponents. Where feasible, use the exponent properties to simplify the terms that contain exponents.
Multiplication and Division. If the equation necessitates it, use multiplication and division to simplify like terms that apply.
Addition and subtraction. Finally, use addition or subtraction the resulting terms of the equation.
Rewrite. Ensure that there are no remaining like terms to simplify, and rewrite the simplified equation.
The Properties For Simplifying Algebraic Expressions
In addition to the PEMDAS sequence, there are a few more rules you must be aware of when simplifying algebraic expressions.
You can only simplify terms with common variables. When adding 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 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 outside of parentheses by distributing them to the terms on the inside, or as follows: a(b+c) = ab + ac.
An extension of the distributive property is referred to as the concept of multiplication. When two stand-alone expressions within parentheses are multiplied, the distribution rule kicks in, and all separate term will will require multiplication by the other terms, making 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 directly outside of an expression in parentheses means 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 on the outside of the parentheses means that it will have distribution applied 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 alter anything when distributed.
How to Simplify Expressions with Exponents
The prior properties were straight-forward enough to implement as they only dealt with properties that affect simple terms with numbers and variables. However, there are additional rules that you have to follow when dealing with exponents and expressions.
Next, we will discuss the laws of exponents. Eight principles influence how we utilize exponentials, which are the following:
Zero Exponent Rule. This principle states that any term with the exponent of 0 is equivalent 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 equivalent variables are apply multiplication, 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 by each other, their quotient subtracts their applicable 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 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 already with an exponent, the term will result in 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 have differing 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.
Simplifying Expressions with the Distributive Property
The distributive property is the principle that says that any term multiplied by an expression within parentheses should be multiplied by all of the expressions within. 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 result is 6x + 10.
Simplifying Expressions with Fractions
Certain expressions contain fractions, and just as with exponents, expressions with fractions also have multiple rules that you have to follow.
When an expression has fractions, here's what to keep in mind.
Distributive property. The distributive property a(b+c) = ab + ac, when applied to fractions, will multiply fractions one at a time by their denominators and numerators.
Laws of exponents. This shows us that fractions will typically 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 expressed in the expression. Apply the PEMDAS rule and ensure that no two terms possess matching variables.
These are the exact principles that you can apply when simplifying any real numbers, whether they are binomials, decimals, square roots, linear equations, quadratic equations, and even logarithms.
Sample Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this example, the rules that must be noted first are the distributive property and the PEMDAS rule. The distributive property will distribute 4 to all the expressions on the inside of the parentheses, while PEMDAS will decide on the order of simplification.
Because of the distributive property, the term outside the parentheses will be multiplied by the terms inside.
4(2x) + 4(5x) + 4(7) - 3y
8x + 20x + 28 - 3y
When simplifying equations, you should add the terms with matching variables, and each term should be in its most simplified 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 states that the the order should start with expressions on the inside of parentheses, and in this example, that expression also necessitates the distributive property. In this scenario, the term y/4 must be distributed within the two terms inside the parentheses, as seen here.
1/3x + y/4(5x) + y/4(2)
Here, let’s set aside the first term for now and simplify the terms with factors assigned 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 to keep things simple as 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 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 means 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 apply simplification to, this becomes our final answer.
Simplifying Expressions FAQs
What should I bear in mind when simplifying expressions?
When simplifying algebraic expressions, remember that you must obey the exponential rule, the distributive property, and PEMDAS rules as well as the concept of multiplication of algebraic expressions. Finally, make sure that every term on your expression is in its most simplified form.
How does solving equations differ from simplifying expressions?
Solving equations and simplifying expressions are vastly different, although, they can be part of the same process the same process due to the fact that you must first simplify expressions before you solve them.
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