Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be challenging for budding students in their primary years of college or even in high school.
However, understanding how to handle these equations is essential because it is basic information that will help them eventually be able to solve higher math and advanced problems across various industries.
This article will share everything you should review to master simplifying expressions. We’ll cover the laws of simplifying expressions and then test our skills via some practice problems.
How Do You Simplify Expressions?
Before learning how to simplify them, you must understand what expressions are in the first place.
In arithmetics, expressions are descriptions that have at least two terms. These terms can contain numbers, variables, or both and can be connected through addition or subtraction.
As an example, let’s review 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 containing coefficients, variables, and occasionally constants, are also referred to as polynomials.
Simplifying expressions is crucial because it paves the way for grasping how to solve them. Expressions can be expressed in complicated ways, and without simplifying them, everyone will have a hard time trying to solve them, with more chance for a mistake.
Of course, every expression differ concerning how they are simplified based on what terms they include, but there are general steps that are applicable to all rational expressions of real numbers, regardless of whether they are square roots, logarithms, or otherwise.
These steps are known as the PEMDAS rule, an abbreviation for parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule states that the order of operations for expressions.
Parentheses. Simplify equations inside the parentheses first by applying addition or subtracting. If there are terms right outside the parentheses, use the distributive property to multiply the term on the outside with the one inside.
Exponents. Where workable, use the exponent rules to simplify the terms that contain exponents.
Multiplication and Division. If the equation necessitates it, use the multiplication and division principles to simplify like terms that are applicable.
Addition and subtraction. Finally, add or subtract the resulting terms of the equation.
Rewrite. Make sure that there are no additional like terms that require simplification, then rewrite the simplified equation.
Here are the Properties For Simplifying Algebraic Expressions
Along with the PEMDAS principle, there are a few more properties you must be aware of when working with algebraic expressions.
You can only apply simplification to 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 expression 8x + 2x can be simplified to 10x by adding coefficients 8 and 2 and keeping the x as it is.
Parentheses that contain another expression directly outside of them need to use 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 distributive rule kicks in, and all individual term will will require multiplication by the other terms, making each set of equations, common factors of each other. Such as is the case here: (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 will also need to have distribution applied, changing the signs of the terms inside the parentheses. Like in this example: -(8x + 2) will turn into -8x - 2.
Likewise, a plus sign outside the parentheses means that it will have distribution applied to the terms on the inside. However, this means that you are able to remove the parentheses and write the expression as is due to the fact that the plus sign doesn’t change anything when distributed.
How to Simplify Expressions with Exponents
The previous principles were simple enough to follow as they only applied to properties 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.
In this section, we will talk about the properties of exponents. 8 properties influence how we process exponents, that includes the following:
Zero Exponent Rule. This property states that any term with a 0 exponent is equivalent 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, 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 applies subtraction to their applicable exponents. This is expressed in 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 result in 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 differing variables will be applied to the appropriate variables, or (ab)m = am * bm.
Power of a Quotient Rule. In fractional exponents, both the numerator and denominator will assume the exponent given, (a/b)m = am/bm.
Simplifying Expressions with the Distributive Property
The distributive property is the principle that shows us that any term multiplied by an expression within parentheses needs be multiplied by all of the expressions 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.
How to Simplify Expressions with Fractions
Certain expressions can consist of fractions, and just like with exponents, expressions with fractions also have multiple rules that you must follow.
When an expression has fractions, here's 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 denominators and numerators.
Laws of exponents. This states 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 should be expressed in the expression. Use the PEMDAS rule and make sure that no two terms contain matching variables.
These are the exact rules 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 example, the rules that must be noted first are PEMDAS and the distributive property. The distributive property will distribute 4 to all the expressions on the inside of the parentheses, while PEMDAS will decide on the order of simplification.
Because of the distributive property, the term outside of 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, you should add all the terms with matching variables, and each 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 expresses that the the order should start with expressions inside parentheses, and in this example, that expression also requires the distributive property. In this example, the term y/4 must be distributed amongst the two terms inside the parentheses, as follows.
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. Remember we know from PEMDAS that fractions will need to multiply their denominators and numerators 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 utilized 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 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 other like terms to apply simplification to, this becomes our final answer.
Simplifying Expressions FAQs
What should I keep in mind when simplifying expressions?
When simplifying algebraic expressions, remember that you must obey the distributive property, PEMDAS, and the exponential rule rules and the concept of multiplication of algebraic expressions. Finally, make sure 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 equations and simplifying expressions are vastly different, but, they can be combined the same process because you have to simplify expressions before you solve them.
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