Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can appear to be challenging for new pupils in their early years of college or even in high school.
However, learning how to deal with these equations is essential because it is primary knowledge that will help them eventually be able to solve higher mathematics and complex problems across various industries.
This article will share everything you should review to master simplifying expressions. We’ll cover the proponents of simplifying expressions and then test our comprehension through some practice problems.
How Do I Simplify an Expression?
Before learning how to simplify expressions, you must grasp what expressions are at their core.
In arithmetics, expressions are descriptions that have no less than two terms. These terms can contain variables, numbers, or both and can be connected through subtraction or addition.
To give an example, let’s review the following expression.
8x + 2y - 3
This expression includes three terms; 8x, 2y, and 3. The first two terms include 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 crucial because it paves the way for grasping how to solve them. Expressions can be written in complicated ways, and without simplifying them, everyone will have a hard time trying to solve them, with more opportunity for error.
Obviously, each expression differ regarding how they're 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 square roots, logarithms, or otherwise.
These steps are called the PEMDAS rule, or parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule shows us the order of operations for expressions.
Parentheses. Simplify 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 outside with the one on the inside.
Exponents. Where possible, use the exponent principles to simplify the terms that contain exponents.
Multiplication and Division. If the equation requires it, use multiplication or division rules to simplify like terms that apply.
Addition and subtraction. Then, use addition or subtraction the resulting terms in the equation.
Rewrite. Ensure that there are no additional like terms to simplify, and then rewrite the simplified equation.
The Properties For Simplifying Algebraic Expressions
Along with the PEMDAS sequence, there are a few more rules you need to 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 expression 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 utilize the distributive property. The distributive property gives you the ability to to simplify terms outside of parentheses by distributing them to the terms on the inside, for example: a(b+c) = ab + ac.
An extension of the distributive property is referred to as the property of multiplication. When two separate expressions within parentheses are multiplied, the distributive principle is applied, and every separate term will need to be multiplied by the other terms, making each set of equations, common factors of each other. Such as is the case here: (a + b)(c + d) = a(c + d) + b(c + d).
A negative sign outside an expression in parentheses means that the negative expression should also need to have distribution applied, changing the signs of the terms on the inside of the parentheses. For example: -(8x + 2) will turn into -8x - 2.
Similarly, a plus sign right outside the parentheses denotes that it will have distribution applied to the terms on the inside. However, this means that you can eliminate 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 previous principles were straight-forward enough to use as they only dealt with principles that impact simple terms with numbers and variables. Despite that, there are more rules that you need to apply when working with expressions with exponents.
Here, we will talk about the properties of exponents. Eight principles impact how we utilize exponents, which are the following:
Zero Exponent Rule. This principle states that any term with a 0 exponent is equal to 1. Or a0 = 1.
Identity Exponent Rule. Any term with the exponent of 1 will not 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 applies subtraction to their respective exponents. This is expressed in the formula am/an = am-n.
Negative Exponents Rule. Any term with a negative exponent equals 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 appropriate variables, or (ab)m = am * bm.
Power of a Quotient Rule. In fractional exponents, both the numerator and denominator will assume the exponent given, (a/b)m = am/bm.
How to Simplify Expressions with the Distributive Property
The distributive property is the rule that says that any term multiplied by an expression on the inside of a 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.
Simplifying Expressions with Fractions
Certain expressions can consist of fractions, and just as with exponents, expressions with fractions also have multiple rules that you have 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 typically 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 should be included in the expression. Use the PEMDAS property and be 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 binomials, decimals, square roots, 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 the expressions inside of the parentheses, while PEMDAS will govern 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, you should add all the terms with the same variables, and each 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 within parentheses, and in this case, 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 seen in this example.
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 attached to them. Since 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 for simplicity 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 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, meaning 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 more like terms to simplify, 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 have to follow PEMDAS, the exponential rule, and the distributive property rules in addition to the concept 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?
Solving equations and simplifying expressions are quite different, although, they can be incorporated into the same process the same process since you have to simplify expressions before solving them.
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