Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions are one of the most challenging for beginner pupils in their first years of college or even in high school.
Still, understanding how to process these equations is essential because it is primary knowledge that will help them eventually be able to solve higher arithmetics and complicated problems across multiple industries.
This article will go over everything you must have to know simplifying expressions. We’ll cover the principles of simplifying expressions and then verify our skills via some sample problems.
How Do I Simplify an Expression?
Before you can be taught how to simplify them, you must learn what expressions are at their core.
In arithmetics, expressions are descriptions that have no less than two terms. These terms can include variables, numbers, or both and can be linked through subtraction or addition.
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 incorporate coefficients, variables, and occasionally constants, are also known as polynomials.
Simplifying expressions is important because it opens up the possibility of learning how to solve them. Expressions can be expressed in complicated ways, and without simplification, you will have a tough time attempting to solve them, with more possibility for error.
Obviously, every expression be different regarding how they are simplified based on what terms they include, but there are typical steps that are applicable 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, an abbreviation for parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule shows us the order of operations for expressions.
Parentheses. Simplify equations inside the parentheses first by adding or subtracting. If there are terms right outside the parentheses, use the distributive property to apply multiplication the term outside with the one on the inside.
Exponents. Where workable, use the exponent rules to simplify the terms that contain exponents.
Multiplication and Division. If the equation calls for it, utilize multiplication and division to simplify like terms that apply.
Addition and subtraction. Lastly, use addition or subtraction the remaining terms of the equation.
Rewrite. Make sure that there are no additional like terms that require simplification, and then rewrite the simplified equation.
Here are the Requirements For Simplifying Algebraic Expressions
Beyond the PEMDAS sequence, there are a few more principles you need to be aware of when working with 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 equation 8x + 2x can be simplified to 10x by adding coefficients 8 and 2 and retaining the variable x as it is.
Parentheses containing 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 on the inside, or as follows: a(b+c) = ab + ac.
An extension of the distributive property is referred to as the principle of multiplication. When two separate expressions within parentheses are multiplied, the distributive principle kicks in, and all separate term will need to be multiplied by the other terms, resulting in each set of equations, common factors of one another. Such as is the case here: (a + b)(c + d) = a(c + d) + b(c + d).
A negative sign directly outside of an expression in parentheses means that the negative expression will also need to have distribution applied, changing the signs of the terms on the inside of the parentheses. For 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 inside. But, this means that you should 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 prior principles were simple enough to follow as they only applied to principles that affect simple terms with numbers and variables. However, there are additional rules that you need to implement when working with exponents and expressions.
In this section, we will review the principles of exponents. Eight properties impact how we process exponentials, which are the following:
Zero Exponent Rule. This property states that any term with the exponent of 0 equals 1. Or a0 = 1.
Identity Exponent Rule. Any term with the exponent of 1 won't alter the value. Or a1 = a.
Product Rule. When two terms with equivalent variables are multiplied by each other, their product will add their two exponents. This is expressed in the formula am × an = am+n
Quotient Rule. When two terms with the same variables are divided, their quotient will subtract 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 result in being the 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 needs to be applied to the respective variables, or (ab)m = am * bm.
Power of a Quotient Rule. In fractional exponents, both the numerator and denominator 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 needs be multiplied by all of the expressions 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 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 have to follow.
When an expression contains 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 states that fractions will usually be the power of the quotient rule, which will subtract the exponents of the numerators and denominators.
Simplification. Only fractions at their lowest should be included in the expression. Apply the PEMDAS property and make sure that no two terms contain matching variables.
These are the exact principles that you can apply when simplifying any real numbers, whether they are square roots, binomials, decimals, logarithms, linear equations, or quadratic equations.
Sample Questions 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 the expressions inside 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 the terms with matching variables, and each term should be in its most simplified 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 expresses that the the order should start with expressions inside parentheses, and in this case, that expression also necessitates the distributive property. In this scenario, the term y/4 must 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. Because 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 for simplicity because 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 each term to one another, 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 simplify, this becomes our final answer.
Simplifying Expressions FAQs
What should I remember when simplifying expressions?
When simplifying algebraic expressions, remember that you must follow the exponential rule, the distributive property, and PEMDAS rules and the principle of multiplication of algebraic expressions. Finally, make sure that every term on your expression is in its most simplified form.
How are simplifying expressions and solving equations different?
Solving and simplifying expressions are very different, but, they can be part of the same process the same process due to the fact that you must first simplify expressions before you begin solving them.
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