Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be intimidating for beginner pupils in their primary years of high school or college.
Nevertheless, learning how to deal with these equations is essential because it is primary knowledge that will help them move on to higher arithmetics and complicated problems across different industries.
This article will go over everything you should review to master simplifying expressions. We’ll cover the principles of simplifying expressions and then validate our skills via some sample problems.
How Does Simplifying Expressions Work?
Before you can be taught how to simplify them, you must grasp what expressions are at their core.
In arithmetics, expressions are descriptions that have a minimum of two terms. These terms can combine numbers, variables, or both and can be linked through addition or subtraction.
To give an example, let’s go over 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 consisting of coefficients, variables, and occasionally constants, are also referred to as polynomials.
Simplifying expressions is crucial because it lays the groundwork for learning how to solve them. Expressions can be written in intricate ways, and without simplification, you will have a hard time trying to solve them, with more chance for a mistake.
Obviously, every expression differ concerning how they are simplified based on what terms they incorporate, 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 refered to as the PEMDAS rule, or parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule declares the order of operations for expressions.
Parentheses. Resolve equations inside the parentheses first by adding or subtracting. 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 contain exponents.
Multiplication and Division. If the equation calls for it, use multiplication and division to simplify like terms that are applicable.
Addition and subtraction. Finally, add or subtract the resulting terms of the equation.
Rewrite. Make sure that there are no more like terms to simplify, and rewrite the simplified equation.
Here are the Requirements For Simplifying Algebraic Expressions
In addition to the PEMDAS principle, there are a few additional rules you need to be aware of when simplifying algebraic expressions.
You can only simplify terms with common variables. When applying addition to these terms, add the coefficient numbers and maintain 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 keeping the x as it is.
Parentheses that contain another expression outside of them need to utilize 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 referred to as the principle of multiplication. When two distinct expressions within parentheses are multiplied, the distribution property is applied, and all individual term will will require multiplication by the other terms, making each set of equations, common factors of each other. Like in this example: (a + b)(c + d) = a(c + d) + b(c + d).
A negative sign directly outside of an expression in parentheses indicates that the negative expression will also need to have distribution applied, changing the signs of the terms inside the parentheses. Like in this example: -(8x + 2) will turn into -8x - 2.
Likewise, a plus sign outside the parentheses will mean 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 owing to the fact that the plus sign doesn’t alter anything when distributed.
How to Simplify Expressions with Exponents
The previous properties were straight-forward enough to follow as they only applied to properties that impact simple terms with variables and numbers. Despite that, there are more rules that you have to implement when dealing with expressions with exponents.
Here, we will review the laws of exponents. 8 principles influence how we utilize exponentials, those are 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 a 1 exponent will not 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 the same variables are divided, their quotient applies subtraction to their two respective exponents. This is expressed in 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 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 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 possess differing variables should be applied to the respective 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.
How to Simplify Expressions with the Distributive Property
The distributive property is the rule that denotes that any term multiplied by an expression within parentheses should be multiplied by all of the expressions within. 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 expression then becomes 6x + 10.
Simplifying Expressions with Fractions
Certain expressions contain fractions, and just like with exponents, expressions with fractions also have several rules that you have to follow.
When an expression includes fractions, here is 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 denominators and numerators.
Laws of exponents. This states that fractions will more likely be the power of the quotient rule, which will subtract the exponents of the numerators and denominators.
Simplification. Only fractions at their lowest form should be written in the expression. Use the PEMDAS principle and ensure that no two terms contain matching variables.
These are the same rules that you can apply when simplifying any real numbers, whether they are binomials, decimals, square roots, linear equations, quadratic equations, and even logarithms.
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 PEMDAS and the distributive property. The distributive property will distribute 4 to all other expressions on the inside of the parentheses, while PEMDAS will govern the order of simplification.
As a result of the distributive property, the term outside of 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, be sure to add the terms with matching variables, and all term should be in its lowest 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 you should begin with expressions inside parentheses, and in this case, that expression also needs the distributive property. In this example, the term y/4 should be distributed to 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 now and simplify the terms with factors attached to them. Since we know from PEMDAS that fractions will require multiplication of their numerators and denominators 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 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 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
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, remember that you must follow the distributive property, PEMDAS, and the exponential rule rules and the principle of multiplication of algebraic expressions. Ultimately, make sure that every term on your expression is in its lowest form.
How are simplifying expressions and solving equations different?
Solving equations and simplifying expressions are quite different, although, they can be incorporated into the same process the same process because you must first simplify expressions before you solve them.
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