Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions are one of the most challenging for new students in their primary years of college or even in high school.
Nevertheless, learning how to deal with these equations is critical because it is foundational information that will help them eventually be able to solve higher mathematics and advanced problems across multiple industries.
This article will go over everything you need to know simplifying expressions. We’ll review the laws of simplifying expressions and then verify our skills through some sample questions.
How Do I Simplify an Expression?
Before learning how to simplify them, you must understand what expressions are to begin with.
In mathematics, expressions are descriptions that have a minimum of two terms. These terms can include variables, numbers, or both and can be linked through addition or subtraction.
To give an example, let’s review the following expression.
8x + 2y - 3
This expression includes three terms; 8x, 2y, and 3. The first two terms contain both numbers (8 and 2) and variables (x and y).
Expressions that incorporate coefficients, variables, and occasionally constants, are also called polynomials.
Simplifying expressions is crucial because it opens up the possibility of understanding how to solve them. Expressions can be expressed in complicated ways, and without simplification, you will have a hard time trying to solve them, with more opportunity for solving them incorrectly.
Of course, all expressions will vary concerning how they are simplified depending on what terms they contain, but there are typical steps that can be applied to all rational expressions of real numbers, whether they are square roots, logarithms, or otherwise.
These steps are called the PEMDAS rule, or parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule states that the order of operations for expressions.
Parentheses. Solve equations between the parentheses first by adding or applying subtraction. If there are terms right outside the parentheses, use the distributive property to apply multiplication the term on the outside with the one inside.
Exponents. Where possible, use the exponent principles to simplify the terms that include exponents.
Multiplication and Division. If the equation necessitates it, use multiplication or division rules to simplify like terms that apply.
Addition and subtraction. Then, add or subtract the remaining terms of the equation.
Rewrite. Ensure that there are no additional like terms to simplify, and rewrite the simplified equation.
The Requirements For Simplifying Algebraic Expressions
Beyond the PEMDAS sequence, there are a few more principles you must be informed of when working 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 equation 8x + 2x can be simplified to 10x by adding coefficients 8 and 2 and leaving the x as it is.
Parentheses that include another expression on the outside of them need to utilize 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 referred to as the concept of multiplication. When two separate expressions within parentheses are multiplied, the distributive principle kicks in, and all separate term will will require multiplication by the other terms, resulting in each set of equations, common factors of each other. Like in this example: (a + b)(c + d) = a(c + d) + b(c + d).
A negative sign directly outside of 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. As is the case in this example: -(8x + 2) will turn into -8x - 2.
Likewise, a plus sign on the outside of the parentheses denotes that it will have distribution applied to the terms inside. Despite that, this means that you are able to eliminate 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 principles were simple enough to use as they only applied to rules that affect simple terms with numbers and variables. However, there are more rules that you need to follow when working with exponents and expressions.
Next, we will review the principles of exponents. 8 principles influence how we utilize exponentials, that includes the following:
Zero Exponent Rule. This principle 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 won't alter the value. Or a1 = a.
Product Rule. When two terms with equivalent variables are multiplied, 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 equals the inverse of that term over 1. This is written as 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 possess unique variables should be applied to the respective variables, or (ab)m = am * bm.
Power of a Quotient Rule. In fractional exponents, both the numerator and denominator will acquire the exponent given, (a/b)m = am/bm.
How to Simplify Expressions with the Distributive Property
The distributive property is the property that shows us that any term multiplied by an expression on the inside of a parentheses should 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 expression then becomes 6x + 10.
Simplifying Expressions with Fractions
Certain expressions contain fractions, and just like with exponents, expressions with fractions also have some rules that you have to follow.
When an expression includes 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 denominators and numerators.
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 property and ensure that no two terms contain the same variables.
These are the exact principles that you can apply when simplifying any real numbers, whether they are binomials, decimals, square roots, linear equations, quadratic equations, and even logarithms.
Practice Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this example, the principles that need to be noted first are the PEMDAS and the distributive property. The distributive property will distribute 4 to all the expressions inside the parentheses, while PEMDAS will govern the order of simplification.
Because of the distributive property, the term on the outside of 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 all the terms with the same variables, and every term should be in its lowest 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 expresses that the you should begin with expressions on the inside of parentheses, and in this scenario, that expression also requires the distributive property. In this example, the term y/4 must be distributed amongst the two terms on the inside of the parentheses, as follows.
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 because 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 means 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
Because there are no remaining 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, keep in mind that you are required to obey PEMDAS, the exponential rule, and the distributive property rules as well as the rule of multiplication of algebraic expressions. Finally, ensure that every term on your expression is in its most simplified form.
How are simplifying expressions and solving equations different?
Solving and simplifying expressions are vastly different, however, they can be incorporated into the same process the same process because you have to simplify expressions before solving them.
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