Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions are one of the most scary for budding pupils in their early years of high school or college.
Nevertheless, grasping how to handle these equations is essential because it is primary information that will help them eventually be able to solve higher math and advanced problems across different industries.
This article will discuss everything you need to learn simplifying expressions. We’ll review the proponents of simplifying expressions and then test what we've learned via some sample problems.
How Does Simplifying Expressions Work?
Before you can be taught how to simplify expressions, you must understand what expressions are in the first place.
In arithmetics, expressions are descriptions that have a minimum of two terms. These terms can combine variables, numbers, or both and can be connected through subtraction or addition.
To give 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 terms consist of both numbers (8 and 2) and variables (x and y).
Expressions containing variables, coefficients, and sometimes constants, are also called polynomials.
Simplifying expressions is crucial because it lays the groundwork for understanding how to solve them. Expressions can be written in convoluted ways, and without simplification, everyone will have a tough time attempting to solve them, with more possibility for solving them incorrectly.
Obviously, each expression differ in how they're simplified depending on what terms they contain, but there are general steps that can be applied to all rational expressions of real numbers, regardless of whether they are logarithms, square roots, etc.
These steps are called the PEMDAS rule, an abbreviation for 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 subtracting. If there are terms right outside the parentheses, use the distributive property to apply multiplication the term 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 the multiplication and division principles to simplify like terms that are applicable.
Addition and subtraction. Lastly, add or subtract the resulting terms in the equation.
Rewrite. Ensure that there are no additional like terms to simplify, and then rewrite the simplified equation.
Here are the Rules For Simplifying Algebraic Expressions
In addition to the PEMDAS rule, there are a few additional rules you must be aware of when simplifying 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 equation 8x + 2x can be simplified to 10x by applying addition to the coefficients 8 and 2 and retaining the variable x as it is.
Parentheses containing another expression directly outside of them need to apply 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 property of multiplication. When two separate expressions within parentheses are multiplied, the distribution principle applies, and all 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 outside an expression in parentheses denotes that the negative expression will 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.
Similarly, a plus sign outside the parentheses will mean that it will have distribution applied to the terms on the inside. But, this means that you should remove 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 straight-forward enough to follow as they only dealt with rules that affect simple terms with variables and numbers. Still, there are additional rules that you have to follow when dealing with exponents and expressions.
Next, we will discuss the properties of exponents. 8 properties impact how we deal with exponentials, that includes 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 won't change in value. Or a1 = a.
Product Rule. When two terms with the same variables are multiplied, 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 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 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 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 required 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 property that says that any term multiplied by an expression within parentheses must be multiplied by all of the expressions on the inside. Let’s watch 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.
Simplifying Expressions with Fractions
Certain expressions can consist of fractions, and just like with exponents, expressions with fractions also have several rules that you have to 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 numerators and denominators.
Laws of exponents. This shows us that fractions will usually be the power of the quotient rule, which will apply subtraction to the exponents of the numerators and denominators.
Simplification. Only fractions at their lowest form should be expressed in the expression. Use the PEMDAS property and make sure that no two terms share matching variables.
These are the same principles that you can apply when simplifying any real numbers, whether they are square roots, binomials, decimals, linear equations, quadratic equations, and even logarithms.
Practice Examples 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 the expressions inside the parentheses, while PEMDAS will govern the order of simplification.
As a result 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, be sure to add all the terms with matching variables, and every term should be in its most simplified 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 on the inside of parentheses, and in this example, that expression also necessitates the distributive property. Here, the term y/4 must be distributed to the two terms inside the parentheses, as follows.
1/3x + y/4(5x) + y/4(2)
Here, let’s put aside the first term for now and simplify the terms with factors associated with 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 every term 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 the same variables multiplied together and multiply their coefficients. This gives us:
24x3 + 4x2 + 18xy + 3y
Since there are no more like terms to apply simplification to, this becomes our final answer.
Simplifying Expressions FAQs
What should I bear in mind when simplifying expressions?
When simplifying algebraic expressions, bear in mind that you are required to follow the exponential rule, the distributive property, and PEMDAS rules as well as the principle of multiplication of algebraic expressions. Ultimately, ensure that every term on your expression is in its lowest form.
How does solving equations differ from simplifying expressions?
Solving equations and simplifying expressions are very different, however, they can be part of the same process the same process since you must first simplify expressions before you begin solving them.
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