Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions are one of the most scary for new pupils in their first years of college or even in high school.
However, grasping how to process these equations is critical because it is primary knowledge that will help them eventually be able to solve higher math and complicated problems across multiple industries.
This article will go over everything you need to know simplifying expressions. We’ll learn the proponents of simplifying expressions and then validate what we've learned through some practice questions.
How Do I Simplify an Expression?
Before you can be taught how to simplify them, you must learn what expressions are to begin with.
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.
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 terms include both numbers (8 and 2) and variables (x and y).
Expressions consisting of coefficients, variables, and occasionally constants, are also known as polynomials.
Simplifying expressions is crucial because it paves the way for understanding how to solve them. Expressions can be written in complicated ways, and without simplifying them, everyone will have a tough time trying to solve them, with more opportunity for error.
Of course, each expression differ concerning how they're simplified depending on what terms they include, but there are common steps that apply to all rational expressions of real numbers, regardless of 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 declares the order of operations for expressions.
Parentheses. Simplify equations between the parentheses first by applying addition or applying subtraction. 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 include exponents.
Multiplication and Division. If the equation calls for it, use multiplication or division rules to simplify like terms that are applicable.
Addition and subtraction. Finally, add or subtract the resulting terms in the equation.
Rewrite. Ensure that there are no additional like terms that need to be simplified, then rewrite the simplified equation.
Here are the Rules For Simplifying Algebraic Expressions
In addition to the PEMDAS sequence, there are a few more rules you need to be aware of when working with algebraic expressions.
You can only apply simplification to terms with common variables. When adding 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 applying addition to the coefficients 8 and 2 and keeping the variable x as it is.
Parentheses containing another expression directly outside of them need to apply the distributive property. The distributive property prompts you to simplify terms on the outside of parentheses by distributing them to the terms inside, or as follows: a(b+c) = ab + ac.
An extension of the distributive property is called the property of multiplication. When two stand-alone expressions within parentheses are multiplied, the distributive principle kicks in, and all unique term will have to be multiplied by the other terms, resulting in 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 directly outside of an expression in parentheses means that the negative expression should also need to be distributed, changing the signs of the terms inside the parentheses. Like in this example: -(8x + 2) will turn into -8x - 2.
Similarly, a plus sign right outside the parentheses will mean that it will have distribution applied to the terms inside. Despite that, this means that you are able to 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 properties were easy enough to use as they only applied to principles that impact simple terms with numbers and variables. However, there are additional rules that you need to implement when dealing with expressions with exponents.
In this section, we will discuss the properties of exponents. Eight principles impact how we utilize exponentials, which 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 won't alter the value. Or a1 = a.
Product Rule. When two terms with the same variables are apply multiplication, their product will add their exponents. This is written as am × an = am+n
Quotient Rule. When two terms with matching variables are divided, their quotient subtracts their applicable exponents. This is written as 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 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 having a 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 unique 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 denotes that any term multiplied by an expression on the inside of a parentheses should be multiplied by all of the expressions on the inside. Let’s witness 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 as with exponents, expressions with fractions also have several rules that you need to follow.
When an expression contains 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 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 numerators and denominators.
Simplification. Only fractions at their lowest should be included in the expression. Use the PEMDAS rule and ensure that no two terms contain the same variables.
These are the same properties that you can apply when simplifying any real numbers, whether they are square roots, binomials, decimals, quadratic equations, logarithms, or linear equations.
Sample Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this case, the properties that should be noted first are the PEMDAS and the distributive property. The distributive property will distribute 4 to the expressions on the inside of the parentheses, while PEMDAS will govern the order of simplification.
As a result 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 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 within parentheses, and in this case, that expression also requires the distributive property. In this example, the term y/4 should be distributed within the two terms on the inside of 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. Since we know from PEMDAS that fractions will require multiplication of their denominators and numerators individually, 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 all terms 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 similar 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 be simplified, this becomes our final answer.
Simplifying Expressions FAQs
What should I bear in mind when simplifying expressions?
When simplifying algebraic expressions, remember that you have to obey PEMDAS, the exponential rule, and the distributive property rules in addition to the principle 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?
Simplifying and solving equations are quite different, however, they can be combined the same process due to the fact that you first need to simplify expressions before you solve them.
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