Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be scary for beginner students in their early years of college or even in high school.
Nevertheless, understanding how to deal with these equations is essential because it is foundational information that will help them move on to higher math and advanced problems across various industries.
This article will share everything you should review to know simplifying expressions. We’ll cover the proponents of simplifying expressions and then validate our comprehension via some sample questions.
How Do You Simplify Expressions?
Before you can learn how to simplify expressions, you must grasp what expressions are at their core.
In mathematics, expressions are descriptions that have no less than two terms. These terms can include variables, numbers, or both and can be connected through subtraction or addition.
For 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 that include variables, coefficients, and sometimes constants, are also referred to as polynomials.
Simplifying expressions is essential because it opens up the possibility of learning how to solve them. Expressions can be written in complicated ways, and without simplification, anyone will have a tough time trying to solve them, with more opportunity for a mistake.
Undoubtedly, every expression vary regarding how they are simplified depending on what terms they incorporate, but there are typical steps that can be applied to all rational expressions of real numbers, regardless of whether they are logarithms, square roots, etc.
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. Solve equations inside the parentheses first by applying addition or using subtraction. If there are terms right outside the parentheses, use the distributive property to multiply the term outside with the one on the inside.
Exponents. Where feasible, use the exponent principles to simplify the terms that include exponents.
Multiplication and Division. If the equation requires it, utilize multiplication and division to simplify like terms that are applicable.
Addition and subtraction. Lastly, add or subtract the resulting terms of the equation.
Rewrite. Make sure that there are no remaining like terms that need to be simplified, and then rewrite the simplified equation.
The Properties For Simplifying Algebraic Expressions
Beyond the PEMDAS principle, there are a few additional principles you should be aware of when simplifying algebraic expressions.
You can only apply simplification to terms with common variables. When adding these terms, add the coefficient numbers and keep 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 keeping the x as it is.
Parentheses that contain another expression on the outside of them need to apply 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 on the inside, as shown here: a(b+c) = ab + ac.
An extension of the distributive property is referred to as the principle of multiplication. When two stand-alone expressions within parentheses are multiplied, the distribution property kicks in, and each individual term will have to be multiplied by the other terms, resulting in each set of equations, common factors of one another. For example: (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 be distributed, changing the signs of the terms inside the parentheses. As is the case in this example: -(8x + 2) will turn into -8x - 2.
Likewise, a plus sign outside the parentheses will mean that it will have distribution applied to the terms inside. But, this means that you can eliminate the parentheses and write the expression as is because the plus sign doesn’t change anything when distributed.
How to Simplify Expressions with Exponents
The prior rules were straight-forward enough to follow as they only dealt with properties that affect simple terms with numbers and variables. Still, there are additional rules that you have to follow when working with expressions with exponents.
Here, we will talk about the principles of exponents. 8 rules influence how we deal with exponentials, which are the following:
Zero Exponent Rule. This rule 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 doesn't alter the value. Or a1 = a.
Product Rule. When two terms with the same variables are multiplied, 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 by each other, their quotient subtracts their applicable 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 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 unique variables needs to be applied to the appropriate 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.
How to Simplify Expressions with the Distributive Property
The distributive property is the principle that says that any term multiplied by an expression on the inside of a parentheses must be multiplied by all of the expressions 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.
Simplifying Expressions with Fractions
Certain expressions contain fractions, and just as with exponents, expressions with fractions also have multiple rules that you have 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 separately by their denominators and numerators.
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 should be expressed in the expression. Apply the PEMDAS rule and make sure that no two terms possess matching variables.
These are the exact rules that you can apply when simplifying any real numbers, whether they are binomials, decimals, square roots, logarithms, linear equations, or quadratic equations.
Sample Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
Here, the principles that need to be noted first are PEMDAS and the distributive property. The distributive property will distribute 4 to the expressions 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 the 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 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 expresses that the you should begin with expressions within parentheses, and in this scenario, that expression also necessitates the distributive property. In this example, the term y/4 will need to be distributed within the two terms within 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 will need to multiply their numerators and denominators separately, we will then have:
y/4 * 5x/1
The expression 5x/1 is used to keep things simple as 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 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, 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
Because there are no more like terms to simplify, this becomes our final answer.
Simplifying Expressions FAQs
What should I keep in mind when simplifying expressions?
When simplifying algebraic expressions, bear in mind that you must follow the distributive property, PEMDAS, and the exponential rule rules in addition to the concept of multiplication of algebraic expressions. In the end, 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?
Solving equations and simplifying expressions are very different, but, they can be incorporated into the same process the same process due to the fact that you must first simplify expressions before solving them.
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