Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be intimidating for beginner learners in their primary years of high school or college.
Nevertheless, understanding how to handle these equations is important because it is primary information that will help them move on to higher arithmetics and advanced problems across multiple industries.
This article will discuss everything you must have to learn simplifying expressions. We’ll review the proponents of simplifying expressions and then test our comprehension through some practice problems.
How Do You Simplify Expressions?
Before you can learn how to simplify them, you must understand what expressions are at their core.
In mathematics, expressions are descriptions that have no less than two terms. These terms can contain 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 contain both numbers (8 and 2) and variables (x and y).
Expressions containing variables, coefficients, and sometimes constants, are also called 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, anyone will have a tough time trying to solve them, with more chance for error.
Obviously, all expressions will be different in how they're simplified depending 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, or parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule shows us the order of operations for expressions.
Parentheses. Resolve equations between the parentheses first by applying addition or using subtraction. 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 principles to simplify the terms that have exponents.
Multiplication and Division. If the equation necessitates it, use multiplication or division rules to simplify like terms that apply.
Addition and subtraction. Then, use addition or subtraction the resulting terms of the equation.
Rewrite. Make sure that there are no remaining like terms that require simplification, and rewrite the simplified equation.
The Requirements For Simplifying Algebraic Expressions
In addition to the PEMDAS principle, there are a few more rules you need to 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 leave the variables as [[is|they are]-70. For example, the equation 8x + 2x can be simplified to 10x by adding coefficients 8 and 2 and retaining the x as it is.
Parentheses that contain another expression directly outside of them need to utilize the distributive property. The distributive property prompts you to simplify terms outside of parentheses by distributing them to the terms on the inside, for example: 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 property is applied, and each unique term will will require multiplication by the other terms, making each set of equations, common factors of each other. For example: (a + b)(c + d) = a(c + d) + b(c + d).
A negative sign outside an expression in parentheses denotes 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 on the outside of the parentheses denotes that it will have distribution applied to the terms inside. However, this means that you should eliminate the parentheses and write the expression as is because the plus sign doesn’t alter anything when distributed.
How to Simplify Expressions with Exponents
The prior properties were simple enough to implement as they only dealt with rules that impact simple terms with numbers and variables. Despite that, there are additional rules that you need to implement when dealing with expressions with exponents.
Next, we will discuss the properties of exponents. 8 principles influence how we deal with exponents, that includes 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 the exponent of 1 doesn't alter the value. Or a1 = a.
Product Rule. When two terms with equivalent variables are multiplied, their product will add their exponents. This is written as am × an = am+n
Quotient Rule. When two terms with matching variables are divided, their quotient applies subtraction to 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 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 result in 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 possess different variables needs to 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 acquire the exponent given, (a/b)m = am/bm.
How to Simplify Expressions with the Distributive Property
The distributive property is the property that denotes that any term multiplied by an expression on the inside of a parentheses needs be multiplied by all of the expressions on the inside. Let’s witness 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 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 some rules that you need to 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 numerators and denominators.
Laws of exponents. This tells us that fractions will usually 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 form should be expressed in the expression. Refer to the PEMDAS principle and ensure that no two terms possess the same variables.
These are the exact rules that you can apply when simplifying any real numbers, whether they are square roots, binomials, decimals, quadratic equations, logarithms, or linear equations.
Practice Examples for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this example, the principles that must 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 dictate the order of simplification.
Because 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 the terms with the same variables, and every term should be in its most simplified 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 case, that expression also requires the distributive property. In this scenario, the term y/4 should be distributed to the two terms on the inside of the parentheses, as follows.
1/3x + y/4(5x) + y/4(2)
Here, let’s put aside the first term for now and simplify the terms with factors assigned to them. Remember 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 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, which means 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 the exponential rule, the distributive property, and PEMDAS rules as well as the rule of multiplication of algebraic expressions. Ultimately, 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 equations and simplifying expressions are very different, although, they can be combined the same process due to the fact that you first need to simplify expressions before solving them.
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