Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions are one of the most scary for budding learners in their first years of high school or college.
Nevertheless, grasping how to process these equations is critical because it is basic knowledge that will help them navigate higher math and complicated problems across multiple industries.
This article will discuss everything you should review to know simplifying expressions. We’ll review the proponents of simplifying expressions and then validate our skills via some sample problems.
How Do I Simplify an Expression?
Before you can learn how to simplify them, you must grasp what expressions are to begin with.
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 review the following expression.
8x + 2y - 3
This expression contains three terms; 8x, 2y, and 3. The first two consist of 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 essential because it lays the groundwork for learning how to solve them. Expressions can be expressed in complicated ways, and without simplification, anyone will have a hard time trying to solve them, with more opportunity for error.
Of course, each expression differ in how they're simplified depending on what terms they include, but there are general steps that can be applied to all rational expressions of real numbers, regardless of whether they are square roots, logarithms, or otherwise.
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 between the parentheses first by adding or using subtraction. If there are terms right outside the parentheses, use the distributive property to multiply the term on the outside with the one inside.
Exponents. Where workable, use the exponent principles 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. Lastly, add or subtract the remaining terms of the equation.
Rewrite. Make sure that there are no more like terms that require simplification, then rewrite the simplified equation.
Here are the Requirements For Simplifying Algebraic Expressions
Beyond the PEMDAS sequence, there are a few additional principles you should be aware 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 expression 8x + 2x can be simplified to 10x by applying addition to the coefficients 8 and 2 and leaving the x as it is.
Parentheses containing another expression directly outside of them need to utilize the distributive property. The distributive property allows you to simplify terms outside of parentheses by distributing them to the terms on the inside, or as follows: a(b+c) = ab + ac.
An extension of the distributive property is referred to as the principle of multiplication. When two distinct expressions within parentheses are multiplied, the distributive rule is applied, and every individual term will need to be multiplied by the other terms, resulting in 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 directly outside of an expression in parentheses means that the negative expression will 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.
Similarly, a plus sign on the outside of the parentheses means that it will have distribution applied to the terms on the inside. But, this means that you should remove the parentheses and write the expression as is because the plus sign doesn’t alter anything when distributed.
How to Simplify Expressions with Exponents
The previous properties were easy enough to follow as they only dealt with principles that affect simple terms with numbers and variables. However, there are more rules that you have to apply when dealing with exponents and expressions.
Next, we will talk about the principles of exponents. 8 rules affect how we deal with exponentials, those are the following:
Zero Exponent Rule. This rule states that any term with a 0 exponent is equivalent to 1. Or a0 = 1.
Identity Exponent Rule. Any term with a 1 exponent won't change in value. Or a1 = a.
Product Rule. When two terms with equivalent 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, their quotient subtracts their respective exponents. This is seen as 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 required variables, or (ab)m = am * bm.
Power of a Quotient Rule. In fractional exponents, both the numerator and denominator will assume the exponent given, (a/b)m = am/bm.
Simplifying Expressions with the Distributive Property
The distributive property is the principle that denotes that any term multiplied by an expression on the inside of a parentheses must be multiplied by all of the expressions within. 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 result is 6x + 10.
How to Simplify Expressions with Fractions
Certain expressions contain fractions, and just as with exponents, expressions with fractions also have several rules that you need to follow.
When an expression contains fractions, here is what to keep in mind.
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 apply subtraction to the exponents of the numerators and denominators.
Simplification. Only fractions at their lowest should be written in the expression. Use the PEMDAS property and ensure that no two terms have matching variables.
These are the same rules that you can apply when simplifying any real numbers, whether they are square roots, binomials, decimals, logarithms, linear equations, or quadratic equations.
Practice Examples 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 on the inside of the parentheses, while PEMDAS will dictate the order of simplification.
Due to the distributive property, the term outside of 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, 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 the order should start with expressions on the inside of parentheses, and in this example, that expression also requires the distributive property. Here, the term y/4 must be distributed to 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 separately, we will then have:
y/4 * 5x/1
The expression 5x/1 is used for simplicity 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 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
Due to the fact that there are no other like terms to apply simplification to, this becomes our final answer.
Simplifying Expressions FAQs
What should I remember when simplifying expressions?
When simplifying algebraic expressions, bear in mind that you are required to obey the distributive property, PEMDAS, and the exponential rule rules and the principle of multiplication of algebraic expressions. Finally, 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 and simplifying expressions are vastly different, but, they can be incorporated into the same process the same process due to the fact that you have to simplify expressions before you solve them.
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