Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be scary for budding students in their first years of college or even in high school.
However, grasping how to deal with these equations is important because it is primary knowledge that will help them navigate higher mathematics and complex problems across multiple industries.
This article will discuss everything you should review to master simplifying expressions. We’ll cover the laws of simplifying expressions and then validate what we've learned with some practice questions.
How Does Simplifying Expressions Work?
Before you can learn how to simplify them, you must understand what expressions are to begin with.
In arithmetics, expressions are descriptions that have no less than two terms. These terms can combine variables, numbers, or both and can be linked through subtraction or addition.
As an example, let’s go over the following expression.
8x + 2y - 3
This expression contains three terms; 8x, 2y, and 3. The first two contain both numbers (8 and 2) and variables (x and y).
Expressions that incorporate variables, coefficients, and sometimes constants, are also referred to as polynomials.
Simplifying expressions is essential because it paves the way for understanding how to solve them. Expressions can be expressed in intricate ways, and without simplifying them, anyone will have a hard time trying to solve them, with more possibility for a mistake.
Of course, all expressions will be different in how they're simplified based on what terms they contain, but there are common steps that are applicable to all rational expressions of real numbers, regardless of whether they are square roots, logarithms, or otherwise.
These steps are called the PEMDAS rule, short for parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule states that the order of operations for expressions.
Parentheses. Resolve equations inside the parentheses first by using addition or using subtraction. If there are terms just outside the parentheses, use the distributive property to multiply the term outside with the one inside.
Exponents. Where feasible, use the exponent rules to simplify the terms that contain exponents.
Multiplication and Division. If the equation requires it, utilize the multiplication and division principles to simplify like terms that apply.
Addition and subtraction. Finally, add or subtract the resulting terms of the equation.
Rewrite. Make sure that there are no additional like terms that need to be simplified, and then rewrite the simplified equation.
The Rules For Simplifying Algebraic Expressions
Beyond the PEMDAS principle, there are a few additional principles you should be informed of when working with 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 expression 8x + 2x can be simplified to 10x by adding coefficients 8 and 2 and retaining the x as it is.
Parentheses that include another expression outside of them need to apply the distributive property. The distributive property gives you the ability to to simplify terms on the outside of parentheses by distributing them to the terms inside, for example: a(b+c) = ab + ac.
An extension of the distributive property is known as the concept of multiplication. When two distinct expressions within parentheses are multiplied, the distributive principle applies, and every separate term will need to be multiplied 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 outside an expression in parentheses means that the negative expression must also need to have distribution applied, changing the signs of the terms on the inside of 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 inside. Despite that, this means that you can eliminate 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 previous rules were easy enough to implement as they only dealt with principles that affect simple terms with variables and numbers. Still, there are more rules that you must follow when working with exponents and expressions.
In this section, we will talk about the laws of exponents. 8 principles affect how we process exponents, that includes the following:
Zero Exponent Rule. This principle states that any term with the exponent of 0 is equal to 1. Or a0 = 1.
Identity Exponent Rule. Any term with the exponent of 1 won't change in value. Or a1 = a.
Product Rule. When two terms with equivalent 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 matching variables are divided by each other, 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 that already has an exponent, the term will end up having a 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 will 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 assume 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 within parentheses must be multiplied by all of the expressions on the inside. Let’s witness 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 expression then becomes 6x + 10.
How to Simplify Expressions with Fractions
Certain expressions contain fractions, and just as with exponents, expressions with fractions also have some rules that you need to follow.
When an expression contains 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 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 denominators and numerators.
Simplification. Only fractions at their lowest should be expressed in the expression. Refer to the PEMDAS principle and ensure that no two terms share matching variables.
These are the exact rules that you can apply when simplifying any real numbers, whether they are decimals, square roots, binomials, quadratic equations, logarithms, or linear equations.
Practice Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this case, the principles that must be noted first are the 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.
Due to 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 all the terms with the same variables, and all term should be in its most simplified 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 expresses that the you should begin with expressions on the inside of parentheses, and in this example, that expression also necessitates the distributive property. Here, the term y/4 will need to 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 assigned to them. Because we know from PEMDAS that fractions will require multiplication of 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 utilized to distribute all terms 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
Due to the fact that there are no other like terms to apply simplification to, this becomes our final answer.
Simplifying Expressions FAQs
What should I keep in mind when simplifying expressions?
When simplifying algebraic expressions, bear in mind that you are required to obey the exponential rule, the distributive property, and PEMDAS rules in addition to the principle of multiplication of algebraic expressions. In the end, make sure that every term on your expression is in its most simplified form.
How does solving equations differ from simplifying expressions?
Solving and simplifying expressions are quite different, however, they can be combined the same process since you first need to simplify expressions before you solve them.
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