Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be scary for new learners in their primary years of college or even in high school.
Nevertheless, learning how to process these equations is critical because it is foundational knowledge that will help them eventually be able to solve higher mathematics and advanced problems across various industries.
This article will share everything you must have to master simplifying expressions. We’ll review the proponents of simplifying expressions and then validate what we've learned via some sample questions.
How Do You Simplify Expressions?
Before learning how to simplify expressions, you must grasp what expressions are to begin with.
In arithmetics, expressions are descriptions that have no less than two terms. These terms can contain variables, numbers, or both and can be linked through subtraction or addition.
For example, let’s go over the following expression.
8x + 2y - 3
This expression contains three terms; 8x, 2y, and 3. The first two contain both numbers (8 and 2) and variables (x and y).
Expressions that incorporate coefficients, variables, and occasionally constants, are also known as polynomials.
Simplifying expressions is important because it lays the groundwork for learning how to solve them. Expressions can be written in convoluted ways, and without simplification, everyone will have a hard time attempting to solve them, with more possibility for error.
Obviously, all expressions will differ regarding how they are simplified based on what terms they contain, but there are common steps that can be applied to all rational expressions of real numbers, whether they are logarithms, square roots, etc.
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 inside the parentheses first by using addition or subtracting. If there are terms right outside the parentheses, use the distributive property to apply multiplication the term on the 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 calls for it, use the multiplication and division principles to simplify like terms that are applicable.
Addition and subtraction. Lastly, add or subtract the remaining terms in the equation.
Rewrite. Ensure that there are no more like terms that require simplification, then rewrite the simplified equation.
Here are the Properties For Simplifying Algebraic Expressions
In addition to the PEMDAS rule, there are a few additional properties you must be informed of when dealing with algebraic expressions.
You can only apply simplification to terms with common variables. When applying addition to 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 x as it is.
Parentheses that contain another expression on the 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, for example: a(b+c) = ab + ac.
An extension of the distributive property is referred to as the concept of multiplication. When two stand-alone expressions within parentheses are multiplied, the distribution rule kicks in, and all individual term will need 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 denotes that the negative expression should also need to be distributed, changing the signs of the terms on the inside of the parentheses. As is the case in this example: -(8x + 2) will turn into -8x - 2.
Similarly, a plus sign on the outside of the parentheses denotes 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 straight-forward enough to use as they only dealt with properties that affect simple terms with numbers and variables. However, there are additional rules that you need to follow when dealing with expressions with exponents.
Next, we will talk about the principles of exponents. Eight rules impact 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 a 1 exponent will not 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 applies subtraction to their two respective exponents. This is written 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 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 that already has an exponent, the term will result in 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 possess differing 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 assume the exponent given, (a/b)m = am/bm.
Simplifying Expressions with the Distributive Property
The distributive property is the rule that states that any term multiplied by an expression within parentheses must be multiplied by all of the expressions on the 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 result is 6x + 10.
Simplifying Expressions with Fractions
Certain expressions can consist of fractions, and just as with exponents, expressions with fractions also have several rules that you need to follow.
When an expression contains 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 denominators and numerators.
Laws of exponents. This shows us 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 state should be written in the expression. Use the PEMDAS principle and be sure that no two terms contain matching variables.
These are the exact rules that you can apply when simplifying any real numbers, whether they are square roots, binomials, decimals, linear equations, quadratic equations, and even logarithms.
Practice Examples for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
Here, the rules that should be noted first are the distributive property and the PEMDAS rule. The distributive property will distribute 4 to all the expressions on the inside of 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 the individual terms inside.
4(2x) + 4(5x) + 4(7) - 3y
8x + 20x + 28 - 3y
When simplifying equations, be sure to add all the terms with matching variables, and all term should be in its most simplified form.
28x + 28 - 3y
Rearrange the equation like this:
28x - 3y + 28
Example 2
Simplify the expression 1/3x + y/4(5x + 2)
The PEMDAS rule expresses that the the order should start with expressions within parentheses, and in this scenario, that expression also requires the distributive property. In this scenario, the term y/4 will need to be distributed within 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 attached to them. Since we know from PEMDAS that fractions will 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 every term 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
Due to the fact that there are no more 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 must obey the exponential rule, the distributive property, and PEMDAS rules in addition to the principle of multiplication of algebraic expressions. In the end, ensure 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 vastly different, but, they can be incorporated into the same process the same process because you have to simplify expressions before solving them.
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