Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be challenging for new pupils in their primary years of college or even in high school.
Nevertheless, understanding how to process these equations is important because it is foundational knowledge that will help them eventually be able to solve higher math and advanced problems across multiple industries.
This article will go over everything you must have to learn simplifying expressions. We’ll review the laws of simplifying expressions and then verify our skills with some practice problems.
How Do I Simplify an Expression?
Before learning how to simplify expressions, you must understand what expressions are to begin with.
In mathematics, 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 combines three terms; 8x, 2y, and 3. The first two contain both numbers (8 and 2) and variables (x and y).
Expressions consisting of coefficients, variables, and occasionally constants, are also called polynomials.
Simplifying expressions is crucial because it lays the groundwork for learning how to solve them. Expressions can be expressed in intricate ways, and without simplification, anyone will have a difficult time attempting to solve them, with more chance for error.
Obviously, all expressions will vary in how they're simplified based on what terms they contain, but there are common steps that apply to all rational expressions of real numbers, regardless of 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 adding or subtracting. If there are terms just outside the parentheses, use the distributive property to multiply the term outside with the one on the inside.
Exponents. Where possible, use the exponent principles to simplify the terms that include exponents.
Multiplication and Division. If the equation calls for it, utilize multiplication and division to simplify like terms that are applicable.
Addition and subtraction. Then, add or subtract the simplified terms of the equation.
Rewrite. Ensure that there are no additional like terms to simplify, then rewrite the simplified equation.
The Rules For Simplifying Algebraic Expressions
Along with the PEMDAS rule, there are a few additional rules you need to be aware of when dealing with algebraic expressions.
You can only simplify terms with common variables. When adding 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 leaving the variable x as it is.
Parentheses containing another expression directly outside of them need to apply the distributive property. The distributive property prompts you 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 called the property of multiplication. When two distinct expressions within parentheses are multiplied, the distributive property applies, and all separate term will need to be multiplied by the other terms, resulting in 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 outside an expression in parentheses denotes that the negative expression should also need to be distributed, changing the signs of the terms inside the parentheses. For example: -(8x + 2) will turn into -8x - 2.
Likewise, a plus sign outside the parentheses means that it will have distribution applied to the terms inside. But, this means that you can remove the parentheses and write the expression as is owing to the fact that the plus sign doesn’t change anything when distributed.
How to Simplify Expressions with Exponents
The previous principles were straight-forward enough to follow as they only dealt with principles that impact simple terms with variables and numbers. However, there are a few other rules that you need to implement when working with expressions with exponents.
Here, we will review the principles of exponents. Eight principles influence how we process exponents, which are the following:
Zero Exponent Rule. This principle states that any term with the exponent of 0 equals 1. Or a0 = 1.
Identity Exponent Rule. Any term with a 1 exponent doesn't alter the value. Or a1 = a.
Product Rule. When two terms with the same variables are multiplied, 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 will subtract their two respective exponents. This is seen as the formula am/an = am-n.
Negative Exponents Rule. Any term with a negative exponent is equal 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 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 possess unique variables needs to be applied to the appropriate variables, or (ab)m = am * bm.
Power of a Quotient Rule. In fractional exponents, both the denominator and numerator will take the exponent given, (a/b)m = am/bm.
Simplifying Expressions with the Distributive Property
The distributive property is the property 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 as with exponents, expressions with fractions also have some rules that you have to follow.
When an expression consist of 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 states 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 form should be expressed in the expression. Refer to the PEMDAS rule and be sure that no two terms possess the same variables.
These are the same principles 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 case, the principles that must be noted first are the PEMDAS and the distributive property. The distributive property will distribute 4 to the expressions inside the parentheses, while PEMDAS will dictate the order of simplification.
As a result 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, be sure to add the terms with the same variables, and all 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 first in order should be expressions inside parentheses, and in this scenario, that expression also needs the distributive property. In this example, the term y/4 will need to be distributed within the two terms on the inside of the parentheses, as seen in this example.
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 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 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, meaning 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
Because there are no remaining 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, keep in mind that you are required to follow PEMDAS, the exponential rule, and the distributive property rules in addition to the principle of multiplication of algebraic expressions. In the end, ensure that every term on your expression is in its lowest form.
How does solving equations differ from simplifying expressions?
Solving equations and simplifying expressions are quite different, although, they can be part of the same process the same process due to the fact that you have to simplify expressions before you begin solving them.
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