Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can appear to be challenging for budding students in their primary years of high school or college.
Still, grasping how to process these equations is important because it is basic knowledge that will help them navigate higher mathematics and complicated problems across various industries.
This article will go over everything you should review to master simplifying expressions. We’ll learn the principles of simplifying expressions and then validate what we've learned with some practice questions.
How Do I Simplify an Expression?
Before you can learn how to simplify them, you must understand what expressions are to begin with.
In arithmetics, expressions are descriptions that have a minimum of two terms. These terms can contain 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 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 coefficients, variables, and occasionally constants, are also called polynomials.
Simplifying expressions is crucial because it paves the way for understanding how to solve them. Expressions can be expressed in complicated ways, and without simplifying them, everyone will have a hard time attempting to solve them, with more opportunity for a mistake.
Undoubtedly, all expressions will vary regarding how they're simplified based on what terms they include, but there are general steps that apply to all rational expressions of real numbers, whether they are square roots, logarithms, or otherwise.
These steps are known as the PEMDAS rule, short for parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule shows us the order of operations for expressions.
Parentheses. Simplify equations inside the parentheses first by applying addition 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 workable, use the exponent rules to simplify the terms that contain exponents.
Multiplication and Division. If the equation requires it, utilize the multiplication and division principles to simplify like terms that are applicable.
Addition and subtraction. Lastly, add or subtract the simplified terms in the equation.
Rewrite. Make sure that there are no remaining like terms to simplify, then rewrite the simplified equation.
The Properties For Simplifying Algebraic Expressions
In addition to the PEMDAS sequence, there are a few more rules you need to 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 leave 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 retaining the variable x as it is.
Parentheses that contain another expression outside of them need to apply the distributive property. The distributive property allows you to simplify terms 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 known as the principle of multiplication. When two separate expressions within parentheses are multiplied, the distribution principle applies, and each individual term will have to be multiplied by the other terms, making 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 will 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.
Similarly, a plus sign right outside the parentheses means 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 due to the fact that the plus sign doesn’t change anything when distributed.
How to Simplify Expressions with Exponents
The previous rules were straight-forward enough to follow as they only applied to principles that affect simple terms with numbers and variables. Despite that, there are a few other rules that you must follow when working with exponents and expressions.
Here, we will review the properties of exponents. Eight principles influence how we utilize exponents, those are the following:
Zero Exponent Rule. This principle states that any term with a 0 exponent 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 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 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 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 needs to be applied to the appropriate 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 property that says 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 result is 6x + 10.
How to Simplify Expressions with Fractions
Certain expressions can consist of fractions, and just like with exponents, expressions with fractions also have some rules that you need to follow.
When an expression consist of fractions, here's 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 numerators and denominators.
Laws of exponents. This tells us that fractions will usually 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. Refer to the PEMDAS rule and make sure that no two terms possess matching variables.
These are the exact properties that you can apply when simplifying any real numbers, whether they are decimals, square roots, binomials, quadratic equations, logarithms, or linear equations.
Sample Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this example, the principles that should be noted first are PEMDAS and the distributive property. The distributive property will distribute 4 to all the expressions inside the parentheses, while PEMDAS will decide on the order of simplification.
As a result of the distributive property, the term on the 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 all the terms with the same 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 states that the the order should start with expressions on the inside of parentheses, and in this case, that expression also necessitates the distributive property. In this example, 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 set aside the first term for the moment and simplify the terms with factors assigned to them. Since we know from PEMDAS that fractions will require multiplication of their numerators and denominators individually, we will then have:
y/4 * 5x/1
The expression 5x/1 is used for simplicity 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 used to distribute every term 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 the same 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 be simplified, this becomes our final answer.
Simplifying Expressions FAQs
What should I remember when simplifying expressions?
When simplifying algebraic expressions, remember that you must obey the exponential rule, the distributive property, and PEMDAS rules as well as the principle of multiplication of algebraic expressions. In the end, make sure that every term on your expression is in its lowest form.
How does solving equations differ from simplifying expressions?
Solving equations and simplifying expressions are quite different, but, they can be combined the same process due to the fact that you first need to simplify expressions before you begin solving them.
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