Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be scary for beginner students in their primary years of college or even in high school.
Still, learning how to process these equations is critical because it is primary knowledge 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 master simplifying expressions. We’ll review the principles of simplifying expressions and then validate our comprehension with some practice questions.
How Does Simplifying Expressions Work?
Before you can learn how to simplify expressions, you must learn what expressions are at their core.
In arithmetics, expressions are descriptions that have no less than two terms. These terms can combine variables, numbers, or both and can be linked 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 contain both numbers (8 and 2) and variables (x and y).
Expressions consisting of variables, coefficients, and sometimes constants, are also referred to as polynomials.
Simplifying expressions is important because it opens up the possibility of grasping how to solve them. Expressions can be written in convoluted ways, and without simplification, anyone will have a hard time attempting to solve them, with more opportunity for error.
Undoubtedly, every expression differ in how they're simplified based on what terms they contain, but there are common steps that can be applied to all rational expressions of real numbers, regardless of whether they are logarithms, square roots, etc.
These steps are refered to as 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 inside the parentheses first by applying addition or applying 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 workable, use the exponent principles to simplify the terms that include 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 remaining terms of the equation.
Rewrite. Make sure that there are no more like terms that need to be simplified, and rewrite the simplified equation.
The Properties For Simplifying Algebraic Expressions
Beyond the PEMDAS rule, there are a few more rules you need to be informed of when dealing with algebraic expressions.
You can only simplify terms with common variables. When adding these terms, add the coefficient numbers and leave 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 directly outside of them need to use the distributive property. The distributive property prompts you 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 referred to as the concept of multiplication. When two stand-alone expressions within parentheses are multiplied, the distribution rule kicks in, and every separate term will need to be multiplied by the other terms, making each set of equations, common factors of one another. Such as is the case here: (a + b)(c + d) = a(c + d) + b(c + d).
A negative sign outside an expression in parentheses denotes that the negative expression must also need to have distribution applied, changing the signs of the terms inside the parentheses. For example: -(8x + 2) will turn into -8x - 2.
Likewise, a plus sign outside the parentheses denotes that it will be distributed to the terms on the 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 implement as they only applied to properties that affect simple terms with numbers and variables. Despite that, there are a few other rules that you must apply when working with exponents and expressions.
In this section, we will review the principles of exponents. Eight properties influence how we process exponentials, that includes the following:
Zero Exponent Rule. This property states that any term with the exponent of 0 is equivalent to 1. Or a0 = 1.
Identity Exponent Rule. Any term with the exponent of 1 doesn't alter the 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 matching variables are divided by each other, their quotient will subtract their respective 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 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 possess differing variables needs to be applied to the respective variables, or (ab)m = am * bm.
Power of a Quotient Rule. In fractional exponents, both the denominator and numerator will take the exponent given, (a/b)m = am/bm.
How to Simplify Expressions with the Distributive Property
The distributive property is the rule that shows us 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 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 resulting expression is 6x + 10.
How to Simplify Expressions with Fractions
Certain expressions contain fractions, and just like with exponents, expressions with fractions also have multiple 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 states that fractions will more likely 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 included in the expression. Apply the PEMDAS rule and make sure that no two terms possess matching variables.
These are the same rules that you can apply when simplifying any real numbers, whether they are binomials, decimals, square roots, quadratic equations, logarithms, or linear equations.
Practice Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this example, the properties that need to be noted first are PEMDAS and the distributive property. The distributive property will distribute 4 to all other expressions inside of 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 the individual terms inside.
4(2x) + 4(5x) + 4(7) - 3y
8x + 20x + 28 - 3y
When simplifying equations, remember to add the terms with matching variables, and all 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 the order should start with expressions within parentheses, and in this case, that expression also necessitates the distributive property. In this example, the term y/4 will need to 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 now and simplify the terms with factors assigned to them. Since we know from PEMDAS that fractions will need to multiply their numerators and denominators individually, 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 used to distribute every 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 similar variables multiplied together and multiply their coefficients. This gives us:
24x3 + 4x2 + 18xy + 3y
Because there are no remaining like terms to be simplified, this becomes our final answer.
Simplifying Expressions FAQs
What should I remember when simplifying expressions?
When simplifying algebraic expressions, keep in mind that you are required to follow PEMDAS, the exponential rule, and the distributive property rules as well as the principle of multiplication of algebraic expressions. In the end, make sure that every term on your expression is in its lowest form.
How does solving equations differ from simplifying expressions?
Simplifying and solving equations are very different, although, they can be combined the same process since you first need to simplify expressions before you begin solving them.
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