Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions are one of the most challenging for new pupils in their early years of college or even in high school.
Nevertheless, understanding how to process these equations is important because it is foundational knowledge that will help them eventually be able to solve higher mathematics and advanced problems across various industries.
This article will discuss everything you need to learn simplifying expressions. We’ll cover the principles of simplifying expressions and then validate our comprehension through some sample problems.
How Does Simplifying Expressions Work?
Before learning how to simplify them, you must understand what expressions are to begin with.
In mathematics, expressions are descriptions that have no less than two terms. These terms can combine 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 contains 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 called polynomials.
Simplifying expressions is essential because it opens up the possibility of understanding how to solve them. Expressions can be written in complicated ways, and without simplifying them, everyone will have a difficult time attempting to solve them, with more opportunity for a mistake.
Undoubtedly, each expression be different in how they're simplified depending on what terms they include, but there are general steps that apply 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 declares the order of operations for expressions.
Parentheses. Solve equations within the parentheses first by using addition or using 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 properties to simplify the terms that have exponents.
Multiplication and Division. If the equation calls for it, utilize the multiplication and division principles to simplify like terms that apply.
Addition and subtraction. Finally, use addition or subtraction the simplified terms of the equation.
Rewrite. Ensure that there are no additional like terms to simplify, and then rewrite the simplified equation.
Here are the Properties For Simplifying Algebraic Expressions
Beyond the PEMDAS sequence, there are a few more rules you need to be informed of when dealing with algebraic expressions.
You can only apply simplification to terms with common variables. When adding these terms, add the coefficient numbers and leave the variables as [[is|they are]-70. For example, the equation 8x + 2x can be simplified to 10x by adding coefficients 8 and 2 and leaving the x as it is.
Parentheses that include another expression directly 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 inside, for example: 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 distribution principle kicks in, and every individual 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 right outside of an expression in parentheses denotes that the negative expression must also need to have distribution applied, changing the signs of the terms on the inside of the parentheses. As is the case in this example: -(8x + 2) will turn into -8x - 2.
Similarly, a plus sign right outside the parentheses denotes that it will have distribution applied to the terms inside. However, this means that you are able to eliminate the parentheses and write the expression as is because the plus sign doesn’t alter anything when distributed.
How to Simplify Expressions with Exponents
The previous principles were straight-forward enough to use as they only applied to rules that impact simple terms with variables and numbers. However, there are a few other rules that you have to apply when dealing with expressions with exponents.
Next, we will review the properties of exponents. Eight properties influence how we deal with exponentials, that includes the following:
Zero Exponent Rule. This property states that any term with the exponent of 0 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 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 written as the formula am/an = am-n.
Negative Exponents Rule. Any term with a negative exponent is equivalent 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 result in 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 will 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.
Simplifying Expressions with the Distributive Property
The distributive property is the property that states that any term multiplied by an expression within parentheses must be multiplied by all of the expressions within. 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 expression then becomes 6x + 10.
Simplifying Expressions with Fractions
Certain expressions contain fractions, and just like with exponents, expressions with fractions also have several rules that you need to follow.
When an expression consist of 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 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 state should be written in the expression. Apply the PEMDAS property and ensure that no two terms share matching variables.
These are the same properties that you can apply when simplifying any real numbers, whether they are decimals, square roots, binomials, logarithms, linear equations, or quadratic equations.
Practice Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
Here, the principles that need to be noted first are the PEMDAS and the distributive property. The distributive property will distribute 4 to all other expressions inside of the parentheses, while PEMDAS will dictate the order of simplification.
Due to the distributive property, the term outside 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 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 inside parentheses, and in this scenario, that expression also needs the distributive property. In this scenario, the term y/4 should be distributed amongst the two terms inside the parentheses, as seen here.
1/3x + y/4(5x) + y/4(2)
Here, let’s put aside the first term for the moment and simplify the terms with factors attached to them. Because we know from PEMDAS that fractions will require multiplication of their denominators and numerators separately, we will then have:
y/4 * 5x/1
The expression 5x/1 is used for simplicity 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 utilized 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 be simplified, this becomes our final answer.
Simplifying Expressions FAQs
What should I bear in mind when simplifying expressions?
When simplifying algebraic expressions, keep in mind that you are required to follow the distributive property, PEMDAS, and the exponential rule 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?
Simplifying and solving equations are vastly different, however, they can be combined the same process because you must first simplify expressions before you solve them.
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