Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be scary for budding students in their primary years of high school or college.
Still, understanding how to process these equations is important because it is basic knowledge that will help them navigate higher math and complicated problems across different industries.
This article will share everything you must have to know simplifying expressions. We’ll cover the proponents of simplifying expressions and then validate our comprehension with some practice problems.
How Does Simplifying Expressions Work?
Before you can be taught how to simplify them, you must grasp what expressions are at their core.
In arithmetics, expressions are descriptions that have a minimum of two terms. These terms can combine numbers, variables, or both and can be linked through subtraction or addition.
As 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 consisting of variables, coefficients, and sometimes constants, are also called polynomials.
Simplifying expressions is crucial because it paves the way for understanding how to solve them. Expressions can be written in convoluted ways, and without simplification, anyone will have a hard time trying to solve them, with more opportunity for error.
Undoubtedly, every expression differ regarding how they're simplified based on what terms they contain, but there are typical steps that are applicable to all rational expressions of real numbers, regardless of whether they are logarithms, square roots, etc.
These steps are known as the PEMDAS rule, or parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule states that the order of operations for expressions.
Parentheses. Resolve equations between the parentheses first by applying addition or using subtraction. If there are terms just outside the parentheses, use the distributive property to multiply the term on the outside with the one inside.
Exponents. Where feasible, use the exponent rules to simplify the terms that contain exponents.
Multiplication and Division. If the equation necessitates it, utilize multiplication or division rules to simplify like terms that apply.
Addition and subtraction. Finally, add or subtract the resulting terms of the equation.
Rewrite. Ensure that there are no remaining like terms that require simplification, then rewrite the simplified equation.
The Properties For Simplifying Algebraic Expressions
Beyond the PEMDAS principle, there are a few additional principles you must be aware of when simplifying algebraic expressions.
You can only simplify 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 adding coefficients 8 and 2 and keeping the x as it is.
Parentheses that contain another expression outside of them need to apply 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 known as the property of multiplication. When two stand-alone expressions within parentheses are multiplied, the distributive rule is applied, and all separate term will need to be multiplied by the other terms, resulting in each set of equations, common factors of each other. Like in this example: (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 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 on the outside of the parentheses means that it will have distribution applied to the terms inside. Despite that, this means that you can remove the parentheses and write the expression as is because 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 principles that affect simple terms with numbers and variables. Despite that, there are a few other rules that you must implement when working with expressions with exponents.
Here, we will talk about the properties of exponents. Eight properties impact how we utilize exponents, that includes the following:
Zero Exponent Rule. This property 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 doesn't change in value. Or a1 = a.
Product Rule. When two terms with equivalent variables are multiplied by each other, 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 by each other, their quotient subtracts their applicable 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 required 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.
Simplifying Expressions with the Distributive Property
The distributive property is the principle 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 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.
Simplifying Expressions with Fractions
Certain expressions contain fractions, and just like with exponents, expressions with fractions also have several rules that you have to follow.
When an expression consist of 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 numerators and denominators.
Laws of exponents. This tells 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 should be expressed in the expression. Apply the PEMDAS rule and be sure that no two terms contain the same variables.
These are the exact properties that you can apply when simplifying any real numbers, whether they are square roots, binomials, decimals, logarithms, linear equations, or quadratic equations.
Practice Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this example, 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 on the inside of the parentheses, while PEMDAS will govern the order of simplification.
Due to the distributive property, the term on the 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 the terms with the same 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 expresses that the first in order should be expressions inside parentheses, and in this example, that expression also requires the distributive property. In this scenario, the term y/4 will need to be distributed within the two terms on the inside of the parentheses, as follows.
1/3x + y/4(5x) + y/4(2)
Here, let’s set aside the first term for the moment and simplify the terms with factors assigned to them. Since 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 for simplicity 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 utilized to distribute each term 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 tells us 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
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 must obey the exponential rule, the distributive property, and PEMDAS rules in addition to the principle of multiplication of algebraic expressions. In the end, make sure that every term on your expression is in its most simplified form.
What is the difference between solving an equation and simplifying an expression?
Simplifying and solving equations are very different, although, they can be combined the same process since you have to simplify expressions before solving them.
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