Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions are one of the most scary for budding students in their early years of college or even in high school.
However, understanding how to process these equations is essential because it is foundational knowledge that will help them navigate higher math and complex problems across various industries.
This article will share everything you must have to master simplifying expressions. We’ll cover the laws of simplifying expressions and then validate what we've learned with some practice problems.
How Do I Simplify an Expression?
Before learning how to simplify them, you must learn what expressions are at their core.
In mathematics, expressions are descriptions that have no less than two terms. These terms can include numbers, variables, or both and can be connected through subtraction or addition.
To give an example, let’s go over the following expression.
8x + 2y - 3
This expression combines three terms; 8x, 2y, and 3. The first two consist of both numbers (8 and 2) and variables (x and y).
Expressions consisting of coefficients, variables, and occasionally constants, are also called polynomials.
Simplifying expressions is important because it lays the groundwork for grasping how to solve them. Expressions can be expressed in complicated ways, and without simplifying them, anyone will have a hard time attempting to solve them, with more opportunity for a mistake.
Of course, all expressions will be different concerning how they're simplified based on what terms they contain, but there are common steps that can be applied 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 shows us the order of operations for expressions.
Parentheses. Simplify equations between the parentheses first by applying addition or subtracting. If there are terms just outside the parentheses, use the distributive property to multiply the term outside with the one on the inside.
Exponents. Where feasible, use the exponent rules to simplify the terms that have 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. Make sure that there are no more like terms that require simplification, then rewrite the simplified equation.
Here are the Rules For Simplifying Algebraic Expressions
Along with the PEMDAS sequence, there are a few additional principles you should be aware of when dealing with algebraic expressions.
You can only apply simplification to terms with common variables. When applying addition to these terms, add the coefficient numbers and keep 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 variable x as it is.
Parentheses that include another expression outside of them need to utilize the distributive property. The distributive property prompts you to simplify terms outside of parentheses by distributing them to the terms inside, for example: a(b+c) = ab + ac.
An extension of the distributive property is called the property of multiplication. When two distinct expressions within parentheses are multiplied, the distributive property applies, and each individual term will will require multiplication by the other terms, making each set of equations, common factors of each other. For example: (a + b)(c + d) = a(c + d) + b(c + d).
A negative sign outside an expression in parentheses indicates that the negative expression should also need to be distributed, changing the signs of the terms inside the parentheses. For example: -(8x + 2) will turn into -8x - 2.
Similarly, a plus sign on the outside of the parentheses means 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 due to the fact that the plus sign doesn’t alter anything when distributed.
How to Simplify Expressions with Exponents
The prior principles were easy enough to implement as they only applied to principles that affect simple terms with numbers and variables. However, there are a few other rules that you need to implement when working with exponents and expressions.
Here, we will talk about the laws of exponents. 8 rules influence how we process exponents, which are the following:
Zero Exponent Rule. This rule states that any term with a 0 exponent is equivalent to 1. Or a0 = 1.
Identity Exponent Rule. Any term with a 1 exponent won't change in value. Or a1 = a.
Product Rule. When two terms with the same 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, their quotient applies subtraction to their two 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 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 result in 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 unique variables will 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 acquire 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 must be multiplied by all of the expressions inside. 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 result is 6x + 10.
Simplifying Expressions with Fractions
Certain expressions contain fractions, and just like with exponents, expressions with fractions also have some rules that you need to follow.
When an expression consist of 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 denominators and numerators.
Laws of exponents. This states that fractions will more likely be the power of the quotient rule, which will apply subtraction to the exponents of the numerators and denominators.
Simplification. Only fractions at their lowest state should be written in the expression. Refer to the PEMDAS rule and make sure that no two terms possess matching variables.
These are the exact rules 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.
Here, the properties that should be noted first are the distributive property and the PEMDAS rule. The distributive property will distribute 4 to all the expressions inside the parentheses, while PEMDAS will decide on the order of simplification.
As a result of the distributive property, the term on the 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, remember to add all 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 scenario, that expression also requires the distributive property. Here, the term y/4 should be distributed within the two terms inside 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 assigned to them. Because we know from PEMDAS that fractions require multiplication of their numerators and denominators 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, meaning 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 simplify, this becomes our final answer.
Simplifying Expressions FAQs
What should I keep in mind when simplifying expressions?
When simplifying algebraic expressions, keep in mind that you have to follow the exponential rule, the distributive property, and PEMDAS rules in addition to the rule of multiplication of algebraic expressions. Finally, ensure that every term on your expression is in its lowest form.
How does solving equations differ from simplifying expressions?
Solving and simplifying expressions are vastly different, however, they can be incorporated into the same process the same process since you must first simplify expressions before you begin solving them.
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