Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can appear to be challenging for beginner learners in their primary years of high school or college.
Nevertheless, understanding how to process these equations is important because it is basic knowledge that will help them navigate higher math and advanced problems across different industries.
This article will go over everything you should review to know simplifying expressions. We’ll cover the proponents of simplifying expressions and then validate our comprehension via some practice problems.
How Do I Simplify an Expression?
Before learning how to simplify them, you must learn what expressions are in the first place.
In mathematics, expressions are descriptions that have at least two terms. These terms can combine numbers, variables, or both and can be linked 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 include both numbers (8 and 2) and variables (x and y).
Expressions that include coefficients, variables, and occasionally constants, are also called polynomials.
Simplifying expressions is essential because it paves the way for learning how to solve them. Expressions can be expressed in convoluted ways, and without simplifying them, anyone will have a hard time attempting to solve them, with more opportunity for error.
Obviously, each expression be different regarding how they're simplified based on what terms they contain, but there are common steps that apply to all rational expressions of real numbers, whether they are logarithms, square roots, etc.
These steps are known as the PEMDAS rule, or parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule declares the order of operations for expressions.
Parentheses. Resolve equations between the parentheses first by applying addition or using subtraction. 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 include exponents.
Multiplication and Division. If the equation requires it, use the multiplication and division principles to simplify like terms that apply.
Addition and subtraction. Finally, add or subtract the simplified terms of the equation.
Rewrite. Ensure that there are no additional like terms that need to be simplified, and then rewrite the simplified equation.
Here are the Properties For Simplifying Algebraic Expressions
In addition to the PEMDAS principle, there are a few more principles you must be aware of when dealing with algebraic expressions.
You can only simplify terms with common variables. When adding 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 retaining the variable x as it is.
Parentheses that contain another expression on the outside of them need to use the distributive property. The distributive property prompts you to simplify terms outside of parentheses by distributing them to the terms inside, or as follows: a(b+c) = ab + ac.
An extension of the distributive property is known as the concept of multiplication. When two stand-alone expressions within parentheses are multiplied, the distribution rule is applied, and each separate 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 indicates that the negative expression should 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.
Likewise, a plus sign on the outside of the parentheses means that it will be distributed to the terms on the inside. Despite that, this means that you should 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 prior rules were straight-forward enough to implement as they only dealt with principles that impact simple terms with variables and numbers. Despite that, there are additional rules that you must apply when dealing with exponents and expressions.
Next, we will review the properties of exponents. Eight properties impact how we deal with exponentials, those are 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 won't 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 by each other, their quotient will subtract their two respective exponents. This is expressed in 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 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 having a 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 have differing variables needs to 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 take the exponent given, (a/b)m = am/bm.
How to Simplify Expressions with the Distributive Property
The distributive property is the principle that states that any term multiplied by an expression on the inside of a parentheses needs be multiplied by all of the expressions inside. Let’s witness 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.
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 need to follow.
When an expression includes 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 subtract the exponents of the denominators and numerators.
Simplification. Only fractions at their lowest should be written in the expression. Apply the PEMDAS property and make sure that no two terms contain matching 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 Examples for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this case, the rules that must be noted first are PEMDAS and the distributive property. The distributive property will distribute 4 to all other expressions inside the parentheses, while PEMDAS will decide on the order of simplification.
Because 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 the terms with the same variables, and every term should be in its most simplified form.
28x + 28 - 3y
Rearrange the equation as follows:
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 scenario, that expression also needs the distributive property. Here, the term y/4 will need to be distributed within the two terms inside the parentheses, as seen in this example.
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. Since we know from PEMDAS that fractions will require multiplication of 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 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 means 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 other like terms to simplify, this becomes our final answer.
Simplifying Expressions FAQs
What should I remember when simplifying expressions?
When simplifying algebraic expressions, keep in mind that you have to obey the exponential rule, the distributive property, and PEMDAS rules and the rule of multiplication of algebraic expressions. Ultimately, ensure that every term on your expression is in its most simplified form.
How are simplifying expressions and solving equations different?
Simplifying and solving equations are very different, however, they can be incorporated into the same process the same process because you have to simplify expressions before you solve them.
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