Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be challenging for budding pupils in their primary years of high school or college.
However, learning how to handle these equations is essential because it is basic knowledge that will help them navigate higher mathematics and advanced problems across various industries.
This article will share everything you need to master simplifying expressions. We’ll cover the proponents of simplifying expressions and then test our comprehension via some sample questions.
How Do You Simplify Expressions?
Before you can be taught how to simplify expressions, you must learn what expressions are to begin with.
In arithmetics, expressions are descriptions that have at least two terms. These terms can include numbers, variables, or both and can be linked through addition or subtraction.
To give an example, let’s go over the following expression.
8x + 2y - 3
This expression includes three terms; 8x, 2y, and 3. The first two terms contain both numbers (8 and 2) and variables (x and y).
Expressions that include coefficients, variables, and occasionally constants, are also known as polynomials.
Simplifying expressions is crucial because it opens up the possibility of understanding how to solve them. Expressions can be expressed in convoluted ways, and without simplifying them, anyone will have a difficult time attempting to solve them, with more chance for solving them incorrectly.
Undoubtedly, each expression vary in how they are simplified depending on what terms they incorporate, but there are typical 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 declares the order of operations for expressions.
Parentheses. Resolve equations within the parentheses first by adding 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 inside.
Exponents. Where feasible, use the exponent properties to simplify the terms that include exponents.
Multiplication and Division. If the equation calls for it, utilize multiplication or division rules to simplify like terms that are applicable.
Addition and subtraction. Finally, add or subtract the simplified terms of the equation.
Rewrite. Make sure that there are no more like terms that need to be simplified, then rewrite the simplified equation.
The Rules For Simplifying Algebraic Expressions
Beyond the PEMDAS principle, there are a few more principles you must be aware of when dealing with algebraic expressions.
You can only simplify 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 keeping the variable x as it is.
Parentheses that contain another expression directly outside of them need to use the distributive property. The distributive property prompts you to simplify terms 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 called the principle of multiplication. When two stand-alone expressions within parentheses are multiplied, the distribution property applies, and every individual term will need to be multiplied by the other terms, making each set of equations, common factors of one another. For example: (a + b)(c + d) = a(c + d) + b(c + d).
A negative sign right outside of an expression in parentheses means that the negative expression must also need to have distribution applied, 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 on the outside of the parentheses means that it will be distributed to the terms inside. But, this means that you are able to 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 implement as they only dealt with properties that affect simple terms with numbers and variables. Despite that, there are more rules that you have to apply when dealing with expressions with exponents.
In this section, we will review the principles of exponents. Eight principles influence how we utilize exponents, that includes the following:
Zero Exponent Rule. This principle states that any term with a 0 exponent is equivalent to 1. Or a0 = 1.
Identity Exponent Rule. Any term with the exponent of 1 won't change in 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 by each other, their quotient will subtract 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 already with an exponent, the term will result in 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 have different variables will be applied to the appropriate variables, or (ab)m = am * bm.
Power of a Quotient Rule. In fractional exponents, both the denominator and numerator will take 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 within parentheses needs 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 expression then becomes 6x + 10.
Simplifying Expressions with Fractions
Certain expressions can consist of fractions, and just as with exponents, expressions with fractions also have multiple rules that you need to follow.
When an expression has fractions, here is 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 denominators and numerators.
Laws of exponents. This tells us that fractions will more likely be the power of the quotient rule, which will apply subtraction to the exponents of the denominators and numerators.
Simplification. Only fractions at their lowest form should be expressed in the expression. Use the PEMDAS property and make sure that no two terms have the same 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 example, the principles that need to be noted first are the PEMDAS and the distributive property. 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 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, be sure to 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 expresses that the the order should start with expressions on the inside of parentheses, and in this case, that expression also necessitates the distributive property. Here, the term y/4 will need to be distributed amongst the two terms within the parentheses, as seen here.
1/3x + y/4(5x) + y/4(2)
Here, let’s put aside the first term for now and simplify the terms with factors assigned to them. Because we know from PEMDAS that fractions will need to multiply their numerators and denominators individually, we will then have:
y/4 * 5x/1
The expression 5x/1 is used to keep things simple 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 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
Since 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, 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. In the end, ensure that every term on your expression is in its lowest form.
How are simplifying expressions and solving equations different?
Solving and simplifying expressions are quite different, although, they can be incorporated into the same process the same process due to the fact that you first need to simplify expressions before you solve them.
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