Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be scary for new students in their early years of college or even in high school.
Still, grasping how to handle these equations is critical because it is basic knowledge that will help them eventually be able to solve higher math and advanced problems across multiple industries.
This article will share everything you need to know simplifying expressions. We’ll review the proponents of simplifying expressions and then test our comprehension with some practice problems.
How Do You Simplify Expressions?
Before learning how to simplify them, you must grasp what expressions are in the first place.
In arithmetics, expressions are descriptions that have no less than two terms. These terms can contain numbers, variables, or both and can be linked through subtraction or addition.
For example, let’s take a look at the following expression.
8x + 2y - 3
This expression includes three terms; 8x, 2y, and 3. The first two consist of both numbers (8 and 2) and variables (x and y).
Expressions containing coefficients, variables, and occasionally constants, are also called polynomials.
Simplifying expressions is crucial because it paves the way for learning how to solve them. Expressions can be written in complicated ways, and without simplification, everyone will have a hard time trying to solve them, with more chance for solving them incorrectly.
Obviously, all expressions will be different regarding how they are simplified based on what terms they contain, but there are common steps that apply to all rational expressions of real numbers, regardless of whether they are logarithms, square roots, etc.
These steps are refered to as the PEMDAS rule, or parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule shows us the order of operations for expressions.
Parentheses. Solve equations within the parentheses first by applying addition or using subtraction. If there are terms just outside the parentheses, use the distributive property to multiply the term outside with the one on the inside.
Exponents. Where workable, use the exponent properties to simplify the terms that have exponents.
Multiplication and Division. If the equation calls for it, utilize multiplication or division rules to simplify like terms that apply.
Addition and subtraction. Lastly, add or subtract the simplified terms of the equation.
Rewrite. Ensure that there are no more like terms that require simplification, and then rewrite the simplified equation.
Here are the Requirements For Simplifying Algebraic Expressions
Beyond the PEMDAS sequence, there are a few additional properties you must be informed of when working 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 expression 8x + 2x can be simplified to 10x by applying addition to the coefficients 8 and 2 and retaining the x as it is.
Parentheses containing another expression on the outside of them need to use the distributive property. The distributive property allows you to simplify terms outside of parentheses by distributing them to the terms inside, or as follows: a(b+c) = ab + ac.
An extension of the distributive property is called the concept of multiplication. When two stand-alone expressions within parentheses are multiplied, the distribution property is applied, and every separate term will will require multiplication by the other terms, making 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 right outside of an expression in parentheses indicates that the negative expression will 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.
Likewise, a plus sign outside the parentheses will mean that it will have distribution applied to the terms on the inside. Despite that, 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 principles were simple enough to follow as they only applied to properties that impact simple terms with numbers and variables. Despite that, there are additional rules that you need to follow when working with exponents and expressions.
In this section, we will review the laws of exponents. 8 rules impact how we utilize 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 won't 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 exponents. This is written as am × an = am+n
Quotient Rule. When two terms with the same variables are divided, their quotient subtracts their respective 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 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 applied to it, or (am)n = amn.
Power of a Product Rule. An exponent applied to two terms that possess differing variables will 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.
How to Simplify Expressions with the Distributive Property
The distributive property is the rule that says that any term multiplied by an expression within parentheses must be multiplied by all of the expressions inside. Let’s see the distributive property applied 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 like with exponents, expressions with fractions also have some rules that you have to follow.
When an expression has 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 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 principle and be sure that no two terms possess the same variables.
These are the exact principles that you can apply when simplifying any real numbers, whether they are decimals, square roots, binomials, logarithms, linear equations, or quadratic equations.
Sample Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
Here, the properties that must be noted first are PEMDAS and the distributive property. The distributive property will distribute 4 to all the expressions inside of the parentheses, while PEMDAS will decide on the order of simplification.
Due to 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 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 first in order should be expressions on the inside of parentheses, and in this case, that expression also requires the distributive property. Here, the term y/4 should be distributed amongst the two terms within the parentheses, as seen here.
1/3x + y/4(5x) + y/4(2)
Here, let’s set aside the first term for now and simplify the terms with factors associated with them. Remember 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 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 each 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 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, remember that you are required to follow the distributive property, PEMDAS, and the exponential rule rules as well as the principle of multiplication of algebraic expressions. In the end, ensure that every term on your expression is in its lowest form.
What is the difference between solving an equation and simplifying an expression?
Solving equations and simplifying expressions are vastly different, although, they can be part of the same process the same process since you first need to simplify expressions before you begin solving them.
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