Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be challenging for beginner pupils in their first years of college or even in high school.
However, learning how to process these equations is essential because it is basic knowledge that will help them navigate higher arithmetics and complex problems across multiple industries.
This article will discuss everything you must have to learn simplifying expressions. We’ll review the principles of simplifying expressions and then validate our comprehension through some sample questions.
How Does Simplifying Expressions Work?
Before you can be taught how to simplify expressions, you must grasp what expressions are in the first place.
In arithmetics, expressions are descriptions that have no less than two terms. These terms can include numbers, variables, or both and can be linked through addition or subtraction.
As an example, let’s take a look at the following expression.
8x + 2y - 3
This expression combines 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 known as polynomials.
Simplifying expressions is essential because it paves the way for grasping how to solve them. Expressions can be written in complicated ways, and without simplification, you will have a tough time trying to solve them, with more possibility for error.
Undoubtedly, each expression be different concerning how they are simplified depending on what terms they include, but there are general steps that apply to all rational expressions of real numbers, whether they are logarithms, square roots, etc.
These steps are refered to as the PEMDAS rule, or parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule shows us the order of operations for expressions.
Parentheses. Resolve equations between the parentheses first by applying addition or using subtraction. If there are terms just 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 include exponents.
Multiplication and Division. If the equation requires it, utilize multiplication or division rules to simplify like terms that are applicable.
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 rules you must be informed of when dealing with algebraic expressions.
You can only simplify terms with common variables. When applying addition to these terms, add the coefficient numbers and maintain 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 x as it is.
Parentheses containing another expression outside of them need to use 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 referred to as the principle of multiplication. When two separate expressions within parentheses are multiplied, the distributive rule is applied, and all unique 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 directly outside of an expression in parentheses indicates 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.
Likewise, a plus sign right outside the parentheses will mean that it will be distributed to the terms 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 change anything when distributed.
How to Simplify Expressions with Exponents
The prior properties were simple enough to use as they only dealt with principles that affect simple terms with numbers and variables. Still, there are additional rules that you need to apply when working with exponents and expressions.
Next, we will talk about the laws of exponents. Eight principles impact how we deal with exponentials, those are the following:
Zero Exponent Rule. This principle 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 change in 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 equal 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 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 unique variables should be applied to the appropriate variables, or (ab)m = am * bm.
Power of a Quotient Rule. In fractional exponents, both the denominator and numerator will assume the exponent given, (a/b)m = am/bm.
How to Simplify Expressions with the Distributive Property
The distributive property is the rule that says that any term multiplied by an expression on the inside of a parentheses should be multiplied by all of the expressions inside. Let’s see the distributive property in action 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.
Simplifying Expressions with Fractions
Certain expressions can consist of fractions, and just like 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 denominators and numerators.
Laws of exponents. This tells us that fractions will typically be the power of the quotient rule, which will subtract the exponents of the numerators and denominators.
Simplification. Only fractions at their lowest form should be written in the expression. Apply the PEMDAS rule and be sure that no two terms contain the same variables.
These are the same principles 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 rules that should 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 govern the order of simplification.
Because of the distributive property, the term outside 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, remember to add all the terms with matching variables, and every term should be in its lowest form.
28x + 28 - 3y
Rearrange the equation like this:
28x - 3y + 28
Example 2
Simplify the expression 1/3x + y/4(5x + 2)
The PEMDAS rule states that the first in order should be expressions inside parentheses, and in this scenario, that expression also needs the distributive property. In this example, the term y/4 will need to be distributed within the two terms within 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 assigned to them. Remember we know from PEMDAS that fractions will require multiplication of their numerators and denominators 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 the same variables multiplied together and multiply their coefficients. This gives us:
24x3 + 4x2 + 18xy + 3y
Due to the fact that there are no remaining like terms to simplify, this becomes our final answer.
Simplifying Expressions FAQs
What should I remember when simplifying expressions?
When simplifying algebraic expressions, keep in mind that you must obey PEMDAS, the exponential rule, and the distributive property rules as well as the concept of multiplication of algebraic expressions. Finally, ensure that every term on your expression is in its most simplified form.
What is the difference between solving an equation and simplifying an expression?
Solving and simplifying expressions are very different, although, they can be incorporated into the same process the same process because you must first simplify expressions before you solve them.
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