Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions are one of the most intimidating for budding pupils in their early years of high school or college.
However, grasping how to handle these equations is important because it is basic knowledge that will help them move on to higher math and complicated problems across various industries.
This article will go over everything you need to master simplifying expressions. We’ll cover the proponents of simplifying expressions and then validate our comprehension via some practice questions.
How Does Simplifying Expressions Work?
Before you can be taught how to simplify them, you must learn what expressions are at their core.
In mathematics, expressions are descriptions that have no less than two terms. These terms can contain variables, numbers, or both and can be connected through subtraction or addition.
As an example, let’s go over the following expression.
8x + 2y - 3
This expression includes three terms; 8x, 2y, and 3. The first two contain both numbers (8 and 2) and variables (x and y).
Expressions consisting of variables, coefficients, and sometimes constants, are also known as polynomials.
Simplifying expressions is essential because it opens up the possibility of learning how to solve them. Expressions can be expressed in intricate ways, and without simplification, you will have a difficult time trying to solve them, with more opportunity for error.
Obviously, every expression vary in how they're simplified based on what terms they include, but there are common 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 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 between the parentheses first by applying addition or subtracting. If there are terms right outside the parentheses, use the distributive property to apply multiplication the term outside with the one on the inside.
Exponents. Where workable, use the exponent properties to simplify the terms that contain exponents.
Multiplication and Division. If the equation calls for it, utilize multiplication and division to simplify like terms that are applicable.
Addition and subtraction. Finally, add or subtract the remaining terms of the equation.
Rewrite. Make sure that there are no remaining like terms that need to be simplified, and rewrite the simplified equation.
The Rules For Simplifying Algebraic Expressions
Along with the PEMDAS principle, there are a few more principles 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 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 retaining the x as it is.
Parentheses containing another expression outside of them need to apply the distributive property. The distributive property allows you to simplify terms on the 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 concept of multiplication. When two separate expressions within parentheses are multiplied, the distributive property applies, and each separate term will have to be multiplied by the other terms, making each set of equations, common factors of each other. Such as is the case here: (a + b)(c + d) = a(c + d) + b(c + d).
A negative sign outside an expression in parentheses means that the negative expression should also need to be distributed, changing the signs of the terms on the inside of the parentheses. Like in this example: -(8x + 2) will turn into -8x - 2.
Similarly, a plus sign right outside the parentheses will mean that it will have distribution applied to the terms on the inside. However, 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 properties were simple enough to implement as they only applied to principles that impact simple terms with numbers and variables. However, there are more rules that you must implement when dealing with exponents and expressions.
In this section, we will discuss the properties of exponents. 8 properties influence how we utilize exponents, those are the following:
Zero Exponent Rule. This property states that any term with the exponent of 0 equals 1. Or a0 = 1.
Identity Exponent Rule. Any term with a 1 exponent doesn'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 matching variables are divided, their quotient applies subtraction to 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 end up being the 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 different variables needs to 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 assume the exponent given, (a/b)m = am/bm.
How to Simplify Expressions with the Distributive Property
The distributive property is the principle 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 witness 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 contain fractions, and just as with exponents, expressions with fractions also have some rules that you must 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 denominators and numerators.
Laws of exponents. This states that fractions will typically be the power of the quotient rule, which will subtract 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 have matching variables.
These are the same 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 case, the principles that need to be noted first are the PEMDAS and the distributive property. The distributive property will distribute 4 to all other expressions on the inside of the parentheses, while PEMDAS will decide on the order of simplification.
As a result of 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 all the terms with matching variables, and each 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 first in order should be expressions on the inside of parentheses, and in this case, that expression also needs the distributive property. In this example, the term y/4 will need to be distributed to 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 the moment and simplify the terms with factors associated with them. Because we know from PEMDAS that fractions require multiplication of their numerators and denominators individually, 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 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 means 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
Because there are no remaining 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, keep in mind that you are required to follow the exponential rule, the distributive property, and PEMDAS rules and the concept of multiplication of algebraic expressions. Finally, 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 and simplifying expressions are very different, but, they can be part of the same process the same process since you have to simplify expressions before solving them.
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