Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions are one of the most scary for new pupils in their early years of college or even in high school.
However, understanding how to handle these equations is critical because it is foundational knowledge that will help them eventually be able to solve higher math and advanced problems across different industries.
This article will discuss everything you need to know simplifying expressions. We’ll cover the proponents of simplifying expressions and then validate what we've learned through some practice problems.
How Do You Simplify Expressions?
Before you can be taught how to simplify expressions, you must grasp what expressions are in the first place.
In mathematics, expressions are descriptions that have no less than two terms. These terms can combine numbers, variables, or both and can be linked through subtraction or addition.
For example, let’s review the following expression.
8x + 2y - 3
This expression includes three terms; 8x, 2y, and 3. The first two include 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 crucial because it opens up the possibility of understanding how to solve them. Expressions can be expressed in intricate ways, and without simplifying them, anyone will have a tough time trying to solve them, with more opportunity for error.
Obviously, all expressions will vary in how they're simplified depending on what terms they incorporate, but there are common steps that apply to all rational expressions of real numbers, regardless of whether they are square roots, logarithms, or otherwise.
These steps are called the PEMDAS rule, an abbreviation for parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule shows us the order of operations for expressions.
Parentheses. Resolve equations within the parentheses first by adding or using subtraction. If there are terms just outside the parentheses, use the distributive property to multiply the term outside with the one inside.
Exponents. Where feasible, use the exponent principles 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 apply.
Addition and subtraction. Lastly, use addition or subtraction the remaining terms of the equation.
Rewrite. Ensure that there are no more like terms that need to be simplified, then rewrite the simplified equation.
The Rules For Simplifying Algebraic Expressions
In addition to the PEMDAS principle, there are a few more principles you need to be informed of when simplifying algebraic expressions.
You can only apply simplification to terms with common variables. When applying addition to 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 on the outside of them need to utilize the distributive property. The distributive property allows you to simplify terms on the 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 property of multiplication. When two stand-alone expressions within parentheses are multiplied, the distributive rule kicks in, and every separate term will will require multiplication by the other terms, resulting in 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 means that the negative expression must also need to be distributed, changing the signs of the terms inside 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 be distributed 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 principles were easy enough to follow as they only dealt with rules that affect simple terms with numbers and variables. Despite that, there are more rules that you must follow when dealing with exponents and expressions.
In this section, we will discuss the properties of exponents. Eight properties affect how we utilize exponentials, 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 won't alter the value. Or a1 = a.
Product Rule. When two terms with the same variables are apply multiplication, their product will add their two exponents. This is expressed in the formula am × an = am+n
Quotient Rule. When two terms with the same variables are divided by each other, their quotient applies subtraction to their respective exponents. This is written 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 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 that were 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 respective variables, or (ab)m = am * bm.
Power of a Quotient Rule. In fractional exponents, both the numerator and denominator will take the exponent given, (a/b)m = am/bm.
How to Simplify Expressions with the Distributive Property
The distributive property is the rule that states that any term multiplied by an expression on the inside of a parentheses must be multiplied by all of the expressions within. Let’s witness the distributive property used 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.
How to Simplify Expressions with Fractions
Certain expressions can consist of fractions, and just like with exponents, expressions with fractions also have several rules that you must follow.
When an expression has fractions, here's what to remember.
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 shows us that fractions will typically be the power of the quotient rule, which will apply subtraction to the exponents of the numerators and denominators.
Simplification. Only fractions at their lowest should be expressed in the expression. Apply the PEMDAS principle and ensure that no two terms possess matching variables.
These are the same properties that you can apply when simplifying any real numbers, whether they are binomials, decimals, square roots, quadratic equations, logarithms, or linear equations.
Practice Examples for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this example, the properties that need to be noted first are the 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 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 all term should be in its most simplified 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 first in order should be expressions inside parentheses, and in this case, that expression also needs the distributive property. Here, the term y/4 should be distributed to the two terms on the inside of 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 attached to them. Because we know from PEMDAS that fractions will 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 since 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 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 tells us 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
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, bear in mind that you must follow the exponential rule, the distributive property, and PEMDAS rules as well as the concept of multiplication of algebraic expressions. In the end, ensure that every term on your expression is in its most simplified form.
What is the difference between solving an equation and simplifying an expression?
Solving and simplifying expressions are vastly different, although, they can be incorporated into the same process the same process because you first need to simplify expressions before solving them.
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