Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions are one of the most scary 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 move on to higher mathematics and complicated problems across multiple industries.
This article will go over everything you need to learn simplifying expressions. We’ll learn the proponents of simplifying expressions and then verify our comprehension with some sample problems.
How Do You Simplify Expressions?
Before you can be taught how to simplify them, you must grasp what expressions are in the first place.
In arithmetics, expressions are descriptions that have a minimum of two terms. These terms can include 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 combines three terms; 8x, 2y, and 3. The first two include both numbers (8 and 2) and variables (x and y).
Expressions that include variables, coefficients, and sometimes constants, are also called polynomials.
Simplifying expressions is essential because it lays the groundwork for grasping how to solve them. Expressions can be expressed in complicated ways, and without simplifying them, you will have a tough time trying to solve them, with more chance for a mistake.
Obviously, each expression vary in how they are simplified depending on what terms they contain, but there are common steps that apply to all rational expressions of real numbers, regardless of whether they are logarithms, square roots, etc.
These steps are refered to as the PEMDAS rule, an abbreviation for parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule states that the order of operations for expressions.
Parentheses. Resolve equations within the parentheses first by applying addition or applying subtraction. If there are terms just outside the parentheses, use the distributive property to multiply the term outside with the one on the inside.
Exponents. Where feasible, use the exponent rules to simplify the terms that have exponents.
Multiplication and Division. If the equation necessitates it, use multiplication or division rules to simplify like terms that are applicable.
Addition and subtraction. Then, add or subtract the resulting terms of the equation.
Rewrite. Make sure that there are no more like terms that require simplification, and rewrite the simplified equation.
Here are the Requirements For Simplifying Algebraic Expressions
Beyond the PEMDAS sequence, there are a few more rules you should be aware 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 expression 8x + 2x can be simplified to 10x by applying addition to the coefficients 8 and 2 and retaining the x as it is.
Parentheses containing another expression outside of them need to use the distributive property. The distributive property prompts 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 principle of multiplication. When two separate expressions within parentheses are multiplied, the distribution rule applies, and all individual term will will require multiplication by the other terms, resulting in 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 denotes that the negative expression will 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.
Likewise, a plus sign outside the parentheses denotes that it will have distribution applied to the terms on the inside. But, this means that you can eliminate the parentheses and write the expression as is due to the fact that the plus sign doesn’t alter anything when distributed.
How to Simplify Expressions with Exponents
The previous principles were simple enough to follow as they only dealt with principles that impact simple terms with numbers and variables. Despite that, there are additional rules that you need to follow when dealing with exponents and expressions.
In this section, we will talk about the laws of exponents. 8 properties influence how we utilize exponents, that includes the following:
Zero Exponent Rule. This property states that any term with a 0 exponent equals 1. Or a0 = 1.
Identity Exponent Rule. Any term with a 1 exponent won't change in value. Or a1 = a.
Product Rule. When two terms with equivalent variables are apply multiplication, their product will add their exponents. This is written as am × an = am+n
Quotient Rule. When two terms with matching variables are divided by each other, their quotient subtracts their two 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 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 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 have unique variables will be applied to the respective variables, or (ab)m = am * bm.
Power of a Quotient Rule. In fractional exponents, both the denominator and numerator will acquire the exponent given, (a/b)m = am/bm.
Simplifying Expressions with the Distributive Property
The distributive property is the rule that shows us that any term multiplied by an expression within parentheses should be multiplied by all of the expressions on the inside. Let’s watch 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 expression then becomes 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 must follow.
When an expression contains 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 tells us that fractions will more likely 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 form should be included in the expression. Refer to the PEMDAS rule and be sure that no two terms have the same variables.
These are the same principles that you can apply when simplifying any real numbers, whether they are decimals, square roots, binomials, linear equations, quadratic equations, and even logarithms.
Practice Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this case, the properties that should be noted first are the distributive property and the PEMDAS rule. The distributive property will distribute 4 to all the expressions on the inside of the parentheses, while PEMDAS will dictate the order of simplification.
Because of the distributive property, the term outside of the parentheses will be multiplied by the terms inside.
4(2x) + 4(5x) + 4(7) - 3y
8x + 20x + 28 - 3y
When simplifying equations, remember to add all the terms with the same variables, and every 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 you should begin with expressions inside parentheses, and in this case, that expression also necessitates the distributive property. Here, the term y/4 must be distributed to the two terms inside the parentheses, as follows.
1/3x + y/4(5x) + y/4(2)
Here, let’s put aside the first term for now and simplify the terms with factors associated with them. Because we know from PEMDAS that fractions will need to multiply their numerators and denominators separately, 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 utilized 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
Since there are no other 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 are required to follow the exponential rule, the distributive property, and PEMDAS rules in addition to 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?
Simplifying and solving equations are very different, but, they can be part of the same process the same process since you first need to simplify expressions before you begin solving them.
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