Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can appear to be challenging for new pupils in their early years of high school or college.
Nevertheless, understanding how to handle these equations is essential because it is primary knowledge that will help them navigate higher mathematics and complicated problems across various industries.
This article will go over everything you need to know simplifying expressions. We’ll learn the proponents of simplifying expressions and then verify our comprehension with some practice problems.
How Do I Simplify an Expression?
Before you can be taught how to simplify them, you must understand what expressions are at their core.
In mathematics, expressions are descriptions that have at least two terms. These terms can contain numbers, variables, or both and can be connected through addition or subtraction.
For example, let’s take a look at the following expression.
8x + 2y - 3
This expression combines three terms; 8x, 2y, and 3. The first two contain both numbers (8 and 2) and variables (x and y).
Expressions that include variables, coefficients, and sometimes constants, are also known as polynomials.
Simplifying expressions is important because it opens up the possibility of grasping how to solve them. Expressions can be written in convoluted ways, and without simplifying them, anyone will have a difficult time trying to solve them, with more chance for solving them incorrectly.
Obviously, all expressions will differ in how they are simplified based 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 known as the PEMDAS rule, short for parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule states that the order of operations for expressions.
Parentheses. Solve equations inside the parentheses first by applying addition or applying subtraction. If there are terms just outside the parentheses, use the distributive property to apply multiplication the term outside with the one on the inside.
Exponents. Where possible, use the exponent rules to simplify the terms that contain exponents.
Multiplication and Division. If the equation necessitates it, use the multiplication and division principles to simplify like terms that apply.
Addition and subtraction. Lastly, use addition or subtraction the simplified terms of the equation.
Rewrite. Make sure that there are no remaining like terms that require simplification, and then rewrite the simplified equation.
The Rules For Simplifying Algebraic Expressions
In addition to the PEMDAS principle, there are a few more rules you should be informed of when dealing with algebraic expressions.
You can only apply simplification to terms with common variables. When adding 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 applying addition to the coefficients 8 and 2 and retaining the 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, as shown here: a(b+c) = ab + ac.
An extension of the distributive property is called the concept of multiplication. When two separate expressions within parentheses are multiplied, the distribution property kicks in, and all individual term will have to be multiplied 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 directly outside of an expression in parentheses denotes that the negative expression should also need to be distributed, changing the signs of the terms inside the parentheses. For example: -(8x + 2) will turn into -8x - 2.
Likewise, a plus sign outside the parentheses will mean that it will be distributed to the terms inside. But, this means that you should eliminate the parentheses and write the expression as is owing to the fact that the plus sign doesn’t alter anything when distributed.
How to Simplify Expressions with Exponents
The prior rules were simple enough to implement as they only applied to rules that impact simple terms with numbers and variables. Still, there are more rules that you have to follow when working with expressions with exponents.
Next, we will discuss the properties of exponents. Eight rules influence how we utilize exponents, those 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 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, their quotient subtracts 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 different variables will be applied to the required variables, or (ab)m = am * bm.
Power of a Quotient Rule. In fractional exponents, both the denominator and numerator will assume 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 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 result is 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 need to follow.
When an expression includes 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 one at a time by their denominators and numerators.
Laws of exponents. This tells us that fractions will typically be the power of the quotient rule, which will subtract the exponents of the denominators and numerators.
Simplification. Only fractions at their lowest should be expressed in the expression. Apply the PEMDAS rule 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.
Practice Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
Here, the rules that need to be noted first are the distributive property and the PEMDAS rule. The distributive property will distribute 4 to all other expressions inside 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 the individual terms inside.
4(2x) + 4(5x) + 4(7) - 3y
8x + 20x + 28 - 3y
When simplifying equations, be sure to add the terms with matching variables, and all term should be in its lowest 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 necessitates the distributive property. In this scenario, the term y/4 should be distributed to the two terms inside the parentheses, as seen here.
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 associated with them. Because we know from PEMDAS that fractions will need to multiply 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 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 similar variables multiplied together and multiply their coefficients. This gives us:
24x3 + 4x2 + 18xy + 3y
Due to the fact that there are no remaining like terms to be simplified, this becomes our final answer.
Simplifying Expressions FAQs
What should I bear in mind when simplifying expressions?
When simplifying algebraic expressions, keep in mind that you must follow the distributive property, PEMDAS, and the exponential rule rules as well as the rule of multiplication of algebraic expressions. Finally, make sure 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 quite different, but, they can be incorporated into the same process the same process since you must first simplify expressions before you begin solving them.
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