Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be challenging for budding pupils in their early years of college or even in high school.
However, learning how to process these equations is essential because it is primary information that will help them navigate higher arithmetics and complex problems across different industries.
This article will discuss everything you should review to master simplifying expressions. We’ll cover the laws of simplifying expressions and then verify our skills through some practice questions.
How Do I Simplify an Expression?
Before learning how to simplify them, you must understand what expressions are to begin with.
In arithmetics, expressions are descriptions that have a minimum of two terms. These terms can include variables, numbers, or both and can be connected through subtraction or addition.
To give an example, let’s review the following expression.
8x + 2y - 3
This expression combines three terms; 8x, 2y, and 3. The first two terms consist of both numbers (8 and 2) and variables (x and y).
Expressions containing coefficients, variables, and occasionally constants, are also referred to as polynomials.
Simplifying expressions is essential because it opens up the possibility of grasping how to solve them. Expressions can be written in convoluted ways, and without simplifying them, anyone will have a hard time attempting to solve them, with more chance for error.
Of course, every expression differ regarding how they are simplified based on what terms they contain, but there are general steps that apply to all rational expressions of real numbers, whether they are square roots, logarithms, or otherwise.
These steps are called 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 adding or using subtraction. If there are terms just outside the parentheses, use the distributive property to apply multiplication the term on the outside with the one on the inside.
Exponents. Where possible, use the exponent properties to simplify the terms that have exponents.
Multiplication and Division. If the equation calls for it, utilize multiplication or division rules to simplify like terms that are applicable.
Addition and subtraction. Then, use addition or subtraction the remaining terms in the equation.
Rewrite. Make sure that there are no remaining like terms to simplify, and then rewrite the simplified equation.
The Requirements For Simplifying Algebraic Expressions
Along with the PEMDAS principle, there are a few more rules you need to 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 leaving the variable x as it is.
Parentheses containing another expression on the outside of them need to apply 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 referred to as the principle of multiplication. When two distinct expressions within parentheses are multiplied, the distribution principle applies, and each individual term will need to be multiplied by the other terms, making each set of equations, common factors of one another. 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 should also need to be distributed, changing the signs of the terms inside the parentheses. For example: -(8x + 2) will turn into -8x - 2.
Likewise, a plus sign on the outside of the parentheses means that it will have distribution applied to the terms inside. However, this means that you can eliminate 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 simple enough to follow as they only applied to properties that affect simple terms with variables and numbers. Despite that, there are a few other rules that you need to apply when dealing with exponents and expressions.
Next, we will review the properties of exponents. 8 rules influence how we process exponentials, that includes the following:
Zero Exponent Rule. This rule states that any term with the exponent of 0 is equivalent to 1. Or a0 = 1.
Identity Exponent Rule. Any term with a 1 exponent will not change in value. Or a1 = a.
Product Rule. When two terms with equivalent variables are multiplied, their product will add their exponents. This is written as am × an = am+n
Quotient Rule. When two terms with matching variables are divided by each other, their quotient subtracts 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 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 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 possess different variables needs to 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 property that says that any term multiplied by an expression on the inside of a parentheses must be multiplied by all of the expressions on the inside. Let’s see 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 as with exponents, expressions with fractions also have multiple rules that you need to follow.
When an expression consist of 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 tells 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 state should be expressed in the expression. Apply the PEMDAS property and be sure that no two terms possess the same 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.
Practice Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
Here, the properties that should be noted first are the PEMDAS and the distributive property. The distributive property will distribute 4 to all the expressions inside of the parentheses, while PEMDAS will govern the order of simplification.
Due to the distributive property, the term outside of the parentheses will be multiplied by the terms inside.
4(2x) + 4(5x) + 4(7) - 3y
8x + 20x + 28 - 3y
When simplifying equations, remember to add all the terms with the same variables, and each 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 expresses that the first in order should be expressions on the inside of parentheses, and in this example, that expression also necessitates the distributive property. In this scenario, the term y/4 should be distributed to 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 the moment and simplify the terms with factors attached to them. Since we know from PEMDAS that fractions require multiplication of their numerators and denominators individually, 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 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 tells us 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
Since there are no more like terms to simplify, this becomes our final answer.
Simplifying Expressions FAQs
What should I bear in mind when simplifying expressions?
When simplifying algebraic expressions, remember that you must obey the distributive property, PEMDAS, and the exponential rule rules in addition to the principle of multiplication of algebraic expressions. Ultimately, make sure that every term on your expression is in its lowest form.
How are simplifying expressions and solving equations different?
Simplifying and solving equations are very different, although, they can be combined the same process due to the fact that you have to simplify expressions before you solve them.
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