Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be scary for new pupils in their first years of high school or college.
Nevertheless, learning how to handle these equations is essential because it is basic knowledge that will help them navigate higher arithmetics and advanced problems across different 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 with some sample problems.
How Does Simplifying Expressions Work?
Before you can be taught how to simplify expressions, you must learn what expressions are at their core.
In arithmetics, expressions are descriptions that have at least two terms. These terms can combine 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 consist of both numbers (8 and 2) and variables (x and y).
Expressions consisting of variables, coefficients, and sometimes constants, are also called polynomials.
Simplifying expressions is important because it opens up the possibility of understanding how to solve them. Expressions can be written in convoluted ways, and without simplifying them, anyone will have a hard time attempting to solve them, with more chance for solving them incorrectly.
Undoubtedly, all expressions will be different in how they are simplified based on what terms they contain, but there are general steps that can be applied to all rational expressions of real numbers, regardless of whether they are logarithms, square roots, etc.
These steps are called the PEMDAS rule, short for parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule states that the order of operations for expressions.
Parentheses. Resolve equations inside the parentheses first by adding or using subtraction. If there are terms just outside the parentheses, use the distributive property to apply multiplication the term outside with the one inside.
Exponents. Where feasible, use the exponent properties to simplify the terms that have exponents.
Multiplication and Division. If the equation requires it, use multiplication or division rules to simplify like terms that apply.
Addition and subtraction. Lastly, use addition or subtraction the resulting terms in the equation.
Rewrite. Ensure that there are no remaining like terms that need to be simplified, and then rewrite the simplified equation.
Here are the Properties For Simplifying Algebraic Expressions
Beyond the PEMDAS principle, there are a few more rules you should be informed of when simplifying algebraic expressions.
You can only apply simplification to terms with common variables. When adding these terms, add the coefficient numbers and maintain the variables as [[is|they are]-70. For example, the equation 8x + 2x can be simplified to 10x by applying addition to the coefficients 8 and 2 and leaving the variable x as it is.
Parentheses that include another expression outside of them need to utilize the distributive property. The distributive property prompts you to simplify terms outside of parentheses by distributing them to the terms on the inside, for example: a(b+c) = ab + ac.
An extension of the distributive property is called the principle of multiplication. When two distinct expressions within parentheses are multiplied, the distribution rule kicks in, and each unique term will need 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 directly outside of an expression in parentheses denotes that the negative expression will also need to have distribution applied, 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 on the outside of the parentheses will mean that it will be distributed to the terms inside. Despite that, this means that you should eliminate the parentheses and write the expression as is because the plus sign doesn’t change anything when distributed.
How to Simplify Expressions with Exponents
The prior properties were easy enough to implement as they only dealt with rules that affect simple terms with numbers and variables. Still, there are additional rules that you need to follow when working with expressions with exponents.
In this section, we will talk about the principles of exponents. Eight principles influence how we process exponentials, 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 the exponent of 1 will not change in value. Or a1 = a.
Product Rule. When two terms with equivalent variables are multiplied, 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 will subtract their two respective exponents. This is expressed in 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 end up 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 should 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 acquire the exponent given, (a/b)m = am/bm.
How to Simplify Expressions with the Distributive Property
The distributive property is the property that shows us that any term multiplied by an expression within parentheses should be multiplied by all of the expressions within. Let’s witness 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 resulting expression 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 need to follow.
When an expression has fractions, here's 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 form should be included in the expression. Use the PEMDAS principle and make sure that no two terms possess matching variables.
These are the exact rules that you can apply when simplifying any real numbers, whether they are decimals, square roots, binomials, linear equations, quadratic equations, and even logarithms.
Practice Examples for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
Here, the properties that should be noted first are the distributive property and the PEMDAS rule. The distributive property will distribute 4 to all other expressions inside of the parentheses, while PEMDAS will govern the order of simplification.
Because of the distributive property, the term outside 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 each term should be in its lowest form.
28x + 28 - 3y
Rearrange the equation as follows:
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 within parentheses, and in this case, that expression also needs the distributive property. Here, the term y/4 will need to be distributed amongst the two terms inside the parentheses, as seen in this example.
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. Remember we know from PEMDAS that fractions require multiplication of their denominators and numerators separately, we will then have:
y/4 * 5x/1
The expression 5x/1 is used to keep things simple 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 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 more like terms to apply simplification to, this becomes our final answer.
Simplifying Expressions FAQs
What should I bear in mind when simplifying expressions?
When simplifying algebraic expressions, remember that you are required to follow PEMDAS, the exponential rule, and the distributive property rules in addition to the rule of multiplication of algebraic expressions. Finally, make sure that every term on your expression is in its most simplified form.
How does solving equations differ from simplifying expressions?
Simplifying and solving equations are vastly different, but, they can be part of the same process the same process because you first need to simplify expressions before you begin solving them.
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