Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can appear to be scary for beginner students in their first years of high school or college.
However, understanding how to deal with these equations is essential because it is foundational information that will help them navigate higher arithmetics and complicated problems across multiple industries.
This article will go over everything you must have to know simplifying expressions. We’ll learn the proponents of simplifying expressions and then test our comprehension with some sample problems.
How Do I Simplify an Expression?
Before learning how to simplify them, you must grasp what expressions are at their core.
In mathematics, expressions are descriptions that have no less than two terms. These terms can contain variables, numbers, or both and can be connected 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 include 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 opens up the possibility of learning how to solve them. Expressions can be expressed in convoluted ways, and without simplifying them, anyone will have a difficult time trying to solve them, with more chance for solving them incorrectly.
Undoubtedly, each expression vary in how they're simplified depending on what terms they include, but there are general steps that are applicable 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 declares the order of operations for expressions.
Parentheses. Resolve equations within the parentheses first by using addition 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 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 simplified terms of the equation.
Rewrite. Ensure that there are no additional like terms to simplify, then rewrite the simplified equation.
Here are the Rules For Simplifying Algebraic Expressions
Along with the PEMDAS principle, there are a few additional rules you need to be aware of when simplifying algebraic expressions.
You can only apply simplification to terms with common variables. When adding these terms, add the coefficient numbers and keep 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 leaving the variable x as it is.
Parentheses containing another expression directly outside of them need to apply the distributive property. The distributive property gives you the ability to to simplify terms 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 known as the property of multiplication. When two distinct expressions within parentheses are multiplied, the distribution rule applies, and all separate term will need to be multiplied 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 outside an expression in parentheses means that the negative expression must also need to be distributed, changing the signs of the terms on the inside of the parentheses. Like in this example: -(8x + 2) will turn into -8x - 2.
Likewise, a plus sign on the outside of the parentheses denotes that it will be distributed to the terms on the inside. However, this means that you should eliminate 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 rules were easy enough to follow as they only dealt with rules that affect simple terms with numbers and variables. However, there are additional rules that you need to implement when dealing with exponents and expressions.
Here, we will review the laws of exponents. 8 rules affect how we process exponentials, which are the following:
Zero Exponent Rule. This rule 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 will not alter the value. Or a1 = a.
Product Rule. When two terms with equivalent variables are apply multiplication, 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 applies subtraction to their respective exponents. This is seen as the formula am/an = am-n.
Negative Exponents Rule. Any term with a negative exponent equals 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 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 differing variables needs to be applied to the appropriate 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 principle that says that any term multiplied by an expression on the inside of a parentheses should be multiplied by all of the expressions within. Let’s see 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.
How to Simplify Expressions with Fractions
Certain expressions can consist of fractions, and just like with exponents, expressions with fractions also have multiple rules that you need to follow.
When an expression has 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 tells us that fractions will usually 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 included in the expression. Use the PEMDAS principle 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 square roots, binomials, decimals, quadratic equations, logarithms, or linear equations.
Sample Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
Here, the properties that should be noted first are the PEMDAS and the distributive property. The distributive property will distribute 4 to all other expressions on the 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 the terms with the same variables, and all term should be in its most simplified 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 expresses that the the order should start with expressions on the inside of parentheses, and in this example, that expression also necessitates the distributive property. In this example, the term y/4 will need to be distributed amongst 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 put aside the first term for the moment and simplify the terms with factors associated with them. Because we know from PEMDAS that fractions will need to multiply their denominators and numerators separately, we will then have:
y/4 * 5x/1
The expression 5x/1 is used for simplicity 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 like variables multiplied together and multiply their coefficients. This gives us:
24x3 + 4x2 + 18xy + 3y
Since there are no remaining like terms to apply simplification to, this becomes our final answer.
Simplifying Expressions FAQs
What should I remember when simplifying expressions?
When simplifying algebraic expressions, remember that you are required to follow the distributive property, PEMDAS, and the exponential rule rules and the principle of multiplication of algebraic expressions. Finally, ensure that every term on your expression is in its most simplified form.
How are simplifying expressions and solving equations different?
Solving equations and simplifying expressions are quite different, however, they can be incorporated into the same process the same process since you have to simplify expressions before you solve them.
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