Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions are one of the most challenging for budding learners in their early years of college or even in high school.
Nevertheless, grasping how to process these equations is important because it is foundational information that will help them move on to higher math and complicated problems across various industries.
This article will discuss everything you must have to know simplifying expressions. We’ll learn the laws of simplifying expressions and then verify our skills via some sample problems.
How Do You Simplify Expressions?
Before learning how to simplify expressions, you must grasp what expressions are at their core.
In arithmetics, expressions are descriptions that have a minimum of two terms. These terms can include numbers, variables, or both and can be connected through addition or subtraction.
As 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 that include 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 simplification, anyone will have a hard time attempting to solve them, with more opportunity for error.
Obviously, every expression vary in how they are simplified based on what terms they contain, but there are common steps that can be applied to all rational expressions of real numbers, whether they are square roots, logarithms, or otherwise.
These steps are called the PEMDAS rule, or parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule states that the order of operations for expressions.
Parentheses. Solve equations between the parentheses first by using addition 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 properties to simplify the terms that have exponents.
Multiplication and Division. If the equation necessitates it, utilize multiplication and division to simplify like terms that are applicable.
Addition and subtraction. Lastly, add or subtract the simplified terms of the equation.
Rewrite. Ensure that there are no remaining 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 additional rules you must be aware of when working with 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 adding coefficients 8 and 2 and retaining the x as it is.
Parentheses that include another expression outside of them need to use the distributive property. The distributive property allows you to simplify terms outside of parentheses by distributing them to the terms on the inside, as shown here: a(b+c) = ab + ac.
An extension of the distributive property is known as the property of multiplication. When two stand-alone expressions within parentheses are multiplied, the distributive principle is applied, and every individual term will need to be multiplied by the other terms, making each set of equations, common factors of one another. Like in this example: (a + b)(c + d) = a(c + d) + b(c + d).
A negative sign right outside of an expression in parentheses indicates that the negative expression should also need to have distribution applied, changing the signs of the terms on the inside of the parentheses. As is the case in this example: -(8x + 2) will turn into -8x - 2.
Likewise, a plus sign outside the parentheses will mean that it will be distributed 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 change anything when distributed.
How to Simplify Expressions with Exponents
The prior principles were easy enough to implement as they only applied to rules that impact simple terms with numbers and variables. Still, there are a few other rules that you have to follow when dealing with exponents and expressions.
In this section, we will discuss the laws of exponents. Eight principles influence how we process exponents, those are the following:
Zero Exponent Rule. This principle states that any term with the exponent of 0 equals 1. Or a0 = 1.
Identity Exponent Rule. Any term with a 1 exponent doesn'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 matching variables are divided, their quotient subtracts their two respective exponents. This is expressed in the formula am/an = am-n.
Negative Exponents Rule. Any term with a negative exponent is equivalent to 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 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 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 acquire the exponent given, (a/b)m = am/bm.
How to Simplify Expressions with the Distributive Property
The distributive property is the principle that shows us that any term multiplied by an expression on the inside of a parentheses must 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.
How to Simplify Expressions with Fractions
Certain expressions can consist of fractions, and just as with exponents, expressions with fractions also have multiple rules that you have to follow.
When an expression includes fractions, here's 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 typically be the power of the quotient rule, which will subtract the exponents of the denominators and numerators.
Simplification. Only fractions at their lowest should be included in the expression. Apply the PEMDAS principle and be sure that no two terms share matching variables.
These are the exact properties that you can apply when simplifying any real numbers, whether they are decimals, square roots, binomials, linear equations, quadratic equations, and even logarithms.
Sample Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this example, the principles 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 dictate the order of simplification.
Because of the distributive property, the term 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 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 scenario, that expression also requires the distributive property. Here, the term y/4 will need to be distributed within the two terms inside the parentheses, as follows.
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 require multiplication of their numerators and denominators separately, 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 used 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 means that we’ll have to add the exponents of two exponential expressions with similar variables multiplied together and multiply their coefficients. This gives us:
24x3 + 4x2 + 18xy + 3y
Since there are no more like terms to be simplified, this becomes our final answer.
Simplifying Expressions FAQs
What should I remember when simplifying expressions?
When simplifying algebraic expressions, bear in mind that you have to follow PEMDAS, the exponential rule, and the distributive property rules and the principle of multiplication of algebraic expressions. Finally, make sure that every term on your expression is in its lowest form.
What is the difference between solving an equation and simplifying an expression?
Simplifying and solving equations are vastly different, however, they can be part of the same process the same process due to the fact that you first need to simplify expressions before solving them.
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