Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions are one of the most challenging for beginner pupils in their primary years of college or even in high school.
However, grasping how to deal with these equations is essential because it is basic information that will help them move on to higher mathematics and complex problems across various industries.
This article will discuss everything you need to learn simplifying expressions. We’ll review the proponents of simplifying expressions and then test what we've learned via some sample questions.
How Does Simplifying Expressions Work?
Before learning how to simplify expressions, you must grasp what expressions are in the first place.
In arithmetics, expressions are descriptions that have no less than two terms. These terms can include 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 contain both numbers (8 and 2) and variables (x and y).
Expressions that include coefficients, variables, and occasionally constants, are also known as polynomials.
Simplifying expressions is essential because it opens up the possibility of grasping how to solve them. Expressions can be expressed in complicated ways, and without simplifying them, you will have a tough time trying to solve them, with more chance for a mistake.
Of course, each expression differ concerning how they're simplified based on what terms they contain, but there are general steps that can be applied to all rational expressions of real numbers, whether they are logarithms, square roots, etc.
These steps are known as the PEMDAS rule, or 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 adding or applying subtraction. If there are terms right outside the parentheses, use the distributive property to apply multiplication the term on the outside with the one inside.
Exponents. Where possible, use the exponent properties to simplify the terms that contain exponents.
Multiplication and Division. If the equation calls for it, use multiplication or division rules to simplify like terms that are applicable.
Addition and subtraction. Lastly, add or subtract the simplified terms of the equation.
Rewrite. Make sure that there are no remaining like terms that need to be simplified, then rewrite the simplified equation.
Here are the Rules For Simplifying Algebraic Expressions
Along with the PEMDAS rule, there are a few more principles 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 leave the variables as [[is|they are]-70. For example, the expression 8x + 2x can be simplified to 10x by applying addition to the coefficients 8 and 2 and retaining the variable x as it is.
Parentheses that include another expression 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 on the inside, for example: a(b+c) = ab + ac.
An extension of the distributive property is referred to as the principle of multiplication. When two stand-alone expressions within parentheses are multiplied, the distribution principle kicks in, and every separate term will need to be multiplied 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 must also need to have distribution applied, changing the signs of the terms on the inside of the parentheses. Like in this example: -(8x + 2) will turn into -8x - 2.
Similarly, a plus sign outside the parentheses denotes that it will have distribution applied to the terms on the 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 easy enough to follow as they only applied to principles that affect simple terms with numbers and variables. However, there are more rules that you have to apply when dealing with exponents and expressions.
Next, we will talk about the laws of exponents. 8 principles influence how we process exponents, that includes the following:
Zero Exponent Rule. This principle states that any term with a 0 exponent is equal to 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 two exponents. This is expressed in the formula am × an = am+n
Quotient Rule. When two terms with matching variables are divided, their quotient subtracts their two respective exponents. This is written 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 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 different variables will 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 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 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 resulting expression is 6x + 10.
How to Simplify Expressions with Fractions
Certain expressions can consist of fractions, and just as with exponents, expressions with fractions also have multiple rules that you must follow.
When an expression has fractions, here's what to remember.
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 states that fractions will usually be the power of the quotient rule, which will subtract the exponents of the denominators and numerators.
Simplification. Only fractions at their lowest form should be written in the expression. Use the PEMDAS property and make sure that no two terms possess the same variables.
These are the same properties that you can apply when simplifying any real numbers, whether they are binomials, decimals, square roots, 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 rules that need to be noted first are the distributive property and the PEMDAS rule. The distributive property will distribute 4 to all other expressions inside of the parentheses, while PEMDAS will dictate 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, you should add all the terms with matching variables, and all term should be in its lowest 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 amongst 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 now and simplify the terms with factors assigned to them. Since we know from PEMDAS that fractions require multiplication of their denominators and numerators individually, 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 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 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, bear 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. Ultimately, make sure 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 very different, but, they can be part of the same process the same process due to the fact that you have to simplify expressions before you solve them.
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