Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be challenging for budding students in their primary years of high school or college.
However, understanding how to deal with these equations is critical because it is primary information that will help them eventually be able to solve higher arithmetics and complex problems across various industries.
This article will go over everything you should review to master simplifying expressions. We’ll cover the principles of simplifying expressions and then verify what we've learned through some practice problems.
How Do I Simplify an Expression?
Before learning how to simplify expressions, you must grasp what expressions are at their core.
In mathematics, expressions are descriptions that have a minimum of two terms. These terms can combine numbers, variables, or both and can be linked through subtraction or addition.
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 terms 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 crucial because it paves the way for understanding how to solve them. Expressions can be expressed in intricate ways, and without simplifying them, everyone will have a hard time attempting to solve them, with more chance for error.
Of course, each expression be different concerning how they are simplified based on what terms they incorporate, but there are common 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, short for parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule declares the order of operations for expressions.
Parentheses. Simplify equations inside the parentheses first by adding or applying subtraction. If there are terms right outside the parentheses, use the distributive property to multiply the term on the outside with the one on the inside.
Exponents. Where feasible, use the exponent rules to simplify the terms that include exponents.
Multiplication and Division. If the equation requires it, utilize multiplication or division rules to simplify like terms that apply.
Addition and subtraction. Then, use addition or subtraction the resulting terms of the equation.
Rewrite. Ensure that there are no remaining like terms that need to be simplified, then rewrite the simplified equation.
The Rules For Simplifying Algebraic Expressions
In addition to the PEMDAS principle, there are a few more properties you should be aware of when working 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 applying addition to the coefficients 8 and 2 and leaving the variable x as it is.
Parentheses containing another expression outside of them need to apply the distributive property. The distributive property prompts you to simplify terms outside of parentheses by distributing them to the terms inside, as shown here: a(b+c) = ab + ac.
An extension of the distributive property is known as the property of multiplication. When two distinct expressions within parentheses are multiplied, the distribution rule kicks in, and every separate term will have 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 indicates that the negative expression must also need to be distributed, changing the signs of the terms inside the parentheses. Like in this example: -(8x + 2) will turn into -8x - 2.
Likewise, a plus sign right outside the parentheses means that it will be distributed to the terms inside. Despite that, this means that you should eliminate 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 previous rules were simple enough to follow as they only dealt with principles that affect simple terms with numbers and variables. Despite that, there are additional rules that you need to follow when working with expressions with exponents.
Here, we will discuss the properties of exponents. 8 properties influence how we deal with exponentials, those are the following:
Zero Exponent Rule. This property states that any term with the exponent of 0 equals 1. Or a0 = 1.
Identity Exponent Rule. Any term with the exponent of 1 doesn't change in value. Or a1 = a.
Product Rule. When two terms with equivalent variables are apply multiplication, their product will add their exponents. This is written as am × an = am+n
Quotient Rule. When two terms with the same variables are divided, their quotient applies subtraction to their respective exponents. This is seen 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 already with an exponent, the term will end up 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 unique 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 acquire the exponent given, (a/b)m = am/bm.
Simplifying Expressions with the Distributive Property
The distributive property is the rule that denotes that any term multiplied by an expression on the inside of a parentheses must be multiplied by all of the expressions on the inside. Let’s see 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 expression then becomes 6x + 10.
Simplifying Expressions with Fractions
Certain expressions contain fractions, and just like with exponents, expressions with fractions also have multiple rules that you must 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 separately by their numerators and denominators.
Laws of exponents. This shows us that fractions will more likely be the power of the quotient rule, which will subtract the exponents of the denominators and numerators.
Simplification. Only fractions at their lowest state should be written in the expression. Use the PEMDAS principle and be sure that no two terms have matching variables.
These are the exact rules that you can apply when simplifying any real numbers, whether they are decimals, square roots, binomials, linear equations, quadratic equations, and even logarithms.
Sample Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
Here, the properties that need to be noted first are the distributive property and the PEMDAS rule. The distributive property will distribute 4 to all the expressions inside of the parentheses, while PEMDAS will dictate 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, you should add the terms with matching variables, and each term should be in its most simplified form.
28x + 28 - 3y
Rearrange the equation this way:
28x - 3y + 28
Example 2
Simplify the expression 1/3x + y/4(5x + 2)
The PEMDAS rule states that the you should begin with expressions on the inside of parentheses, and in this scenario, that expression also requires the distributive property. Here, the term y/4 should be distributed within 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 assigned to them. Since we know from PEMDAS that fractions 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 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 like variables multiplied together and multiply their coefficients. This gives us:
24x3 + 4x2 + 18xy + 3y
Since there are no remaining like terms to simplify, 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 have to follow PEMDAS, the exponential rule, and the distributive property rules in addition to the concept of multiplication of algebraic expressions. Ultimately, make sure that every term on your expression is in its most simplified form.
How does solving equations differ from simplifying expressions?
Solving equations and simplifying expressions are very different, but, they can be part of the same process the same process because you first need to simplify expressions before you solve them.
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