Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be scary for new pupils in their first years of high school or college.
Still, learning how to handle these equations is important because it is basic knowledge that will help them eventually be able to solve higher mathematics and advanced problems across various industries.
This article will share everything you should review to know simplifying expressions. We’ll learn the principles of simplifying expressions and then verify what we've learned via some practice questions.
How Does Simplifying Expressions Work?
Before you can be taught how to simplify them, you must understand what expressions are in the first place.
In mathematics, expressions are descriptions that have a minimum of two terms. These terms can include variables, numbers, or both and can be connected through addition or subtraction.
For example, let’s go over the following expression.
8x + 2y - 3
This expression contains three terms; 8x, 2y, and 3. The first two consist of 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 crucial because it paves the way for grasping how to solve them. Expressions can be expressed in complicated ways, and without simplifying them, everyone will have a tough time attempting to solve them, with more opportunity for a mistake.
Of course, each expression vary concerning how they are simplified depending on what terms they contain, but there are common steps that can be applied to all rational expressions of real numbers, whether they are logarithms, square roots, etc.
These steps are known as the PEMDAS rule, an abbreviation for parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule states that the order of operations for expressions.
Parentheses. Simplify equations inside the parentheses first by applying addition or subtracting. If there are terms just outside the parentheses, use the distributive property to multiply the term on the outside with the one on the inside.
Exponents. Where possible, use the exponent principles to simplify the terms that contain exponents.
Multiplication and Division. If the equation necessitates it, use multiplication or division rules to simplify like terms that apply.
Addition and subtraction. Then, use addition or subtraction the remaining terms of the equation.
Rewrite. Ensure that there are no additional like terms to simplify, then rewrite the simplified equation.
The Properties For Simplifying Algebraic Expressions
Along with the PEMDAS rule, there are a few additional rules you need to be informed of when simplifying algebraic expressions.
You can only simplify terms with common variables. When adding these terms, add the coefficient numbers and leave the variables as [[is|they are]-70. For example, the expression 8x + 2x can be simplified to 10x by applying addition to the coefficients 8 and 2 and leaving the 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, as shown here: a(b+c) = ab + ac.
An extension of the distributive property is known as the principle of multiplication. When two separate expressions within parentheses are multiplied, the distributive rule applies, and every unique term will need to be multiplied by the other terms, resulting in each set of equations, common factors of each other. For example: (a + b)(c + d) = a(c + d) + b(c + d).
A negative sign directly outside of an expression in parentheses denotes that the negative expression will also need to have distribution applied, 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 will mean that it will have distribution applied to the terms on the inside. Despite that, 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 previous rules were straight-forward enough to implement as they only dealt with rules that impact simple terms with variables and numbers. However, there are more rules that you have to apply when working with expressions with exponents.
Next, we will review the principles of exponents. Eight properties influence how we process exponentials, those are the following:
Zero Exponent Rule. This property states that any term with a 0 exponent equals 1. Or a0 = 1.
Identity Exponent Rule. Any term with the exponent of 1 doesn't alter the value. Or a1 = a.
Product Rule. When two terms with the same 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 applies subtraction to their applicable 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 end up 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 have different variables should be applied to the respective variables, or (ab)m = am * bm.
Power of a Quotient Rule. In fractional exponents, both the denominator and numerator will take 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 result is 6x + 10.
How to Simplify Expressions with Fractions
Certain expressions can consist of fractions, and just as with exponents, expressions with fractions also have some rules that you must follow.
When an expression has fractions, here's what to remember.
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 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 form should be included in the expression. Use the PEMDAS property and make sure that no two terms have the same variables.
These are the same rules 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.
In this case, the rules that need to be noted first are the PEMDAS and the distributive property. The distributive property will distribute 4 to all the expressions on the inside of the parentheses, while PEMDAS will decide on the order of simplification.
Due to 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, be sure to add the terms with the same variables, and all 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 states that the the order should start with expressions within parentheses, and in this example, that expression also needs the distributive property. In this example, the term y/4 must be distributed within 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 the moment and simplify the terms with factors associated with 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 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 utilized 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, meaning 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 other like terms to apply simplification to, 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 must 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?
Solving equations and simplifying expressions are vastly different, although, they can be part of the same process the same process due to the fact that you have to simplify expressions before you solve them.
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