Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be intimidating for budding students in their early years of high school or college.
Nevertheless, understanding how to handle these equations is important because it is basic information that will help them navigate higher mathematics and advanced problems across different industries.
This article will share everything you should review to master simplifying expressions. We’ll review the principles of simplifying expressions and then validate our skills with some sample problems.
How Does Simplifying Expressions Work?
Before you can learn how to simplify expressions, you must learn what expressions are to begin with.
In mathematics, expressions are descriptions that have a minimum of two terms. These terms can combine variables, numbers, or both and can be linked through subtraction or addition.
To give an example, let’s review the following expression.
8x + 2y - 3
This expression includes three terms; 8x, 2y, and 3. The first two include both numbers (8 and 2) and variables (x and y).
Expressions containing coefficients, variables, and occasionally constants, are also known as polynomials.
Simplifying expressions is essential because it opens up the possibility of learning how to solve them. Expressions can be expressed in intricate ways, and without simplification, everyone will have a difficult time trying to solve them, with more possibility for error.
Obviously, all expressions will be different concerning how they're simplified depending on what terms they include, but there are general steps that apply to all rational expressions of real numbers, regardless of whether they are logarithms, square roots, etc.
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 within the parentheses first by adding or using subtraction. If there are terms right outside the parentheses, use the distributive property to apply multiplication the term on the outside with the one on the inside.
Exponents. Where workable, use the exponent principles to simplify the terms that have exponents.
Multiplication and Division. If the equation necessitates it, utilize the multiplication and division principles to simplify like terms that apply.
Addition and subtraction. Then, add or subtract the remaining 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
Beyond the PEMDAS principle, there are a few more principles you should be aware of when simplifying algebraic expressions.
You can only simplify 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 applying addition to the coefficients 8 and 2 and keeping the variable x as it is.
Parentheses that include another expression outside of them need to utilize the distributive property. The distributive property allows you to simplify terms outside of parentheses by distributing them to the terms inside, as shown here: a(b+c) = ab + ac.
An extension of the distributive property is known as the concept of multiplication. When two distinct expressions within parentheses are multiplied, the distributive property kicks in, and all individual term will will require multiplication by the other terms, resulting in 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 denotes that the negative expression must also need to be distributed, 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 on the outside of the parentheses denotes that it will have distribution applied to the terms on the inside. But, this means that you should eliminate 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 rules were straight-forward enough to use as they only dealt with principles that impact simple terms with numbers and variables. However, there are additional rules that you need to apply when working with expressions with exponents.
Next, we will talk about the laws of exponents. Eight principles affect how we utilize exponentials, which 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 doesn't change in 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 the same variables are divided by each other, their quotient will subtract their respective exponents. This is written 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 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 being the 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 different variables will 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 acquire the exponent given, (a/b)m = am/bm.
Simplifying Expressions with the Distributive Property
The distributive property is the rule that says that any term multiplied by an expression within parentheses should be multiplied by all of the expressions inside. Let’s see 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 contain fractions, and just as with exponents, expressions with fractions also have several rules that you need 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 numerators and denominators.
Laws of exponents. This shows 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 expressed in the expression. Apply the PEMDAS property and be sure that no two terms possess matching variables.
These are the exact principles that you can apply when simplifying any real numbers, whether they are square roots, binomials, decimals, linear equations, quadratic equations, and even logarithms.
Practice Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
Here, the rules that must 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 dictate the order of simplification.
Due to 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, you should add all the terms with matching variables, and every 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 you should begin with expressions on the inside of parentheses, and in this scenario, that expression also requires the distributive property. In this scenario, the term y/4 will need to 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 attached to them. Remember we know from PEMDAS that fractions require multiplication of their denominators and numerators separately, we will then have:
y/4 * 5x/1
The expression 5x/1 is used to keep things simple 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 used to distribute every 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 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
Because there are no more like terms to be simplified, this becomes our final answer.
Simplifying Expressions FAQs
What should I keep in mind when simplifying expressions?
When simplifying algebraic expressions, bear in mind that you are required to follow the exponential rule, the distributive property, and PEMDAS rules as well as the concept of multiplication of algebraic expressions. Ultimately, ensure that every term on your expression is in its most simplified form.
How are simplifying expressions and solving equations different?
Simplifying and solving equations are very different, but, they can be incorporated into the same process the same process due to the fact that you first need to simplify expressions before you begin solving them.
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