Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can be scary for budding students in their early years of college or even in high school.
Nevertheless, learning how to handle these equations is critical because it is basic knowledge that will help them move on to higher arithmetics and complex problems across different industries.
This article will share everything you need to master simplifying expressions. We’ll review the principles of simplifying expressions and then verify our comprehension with some sample problems.
How Do I Simplify an Expression?
Before you can be taught how to simplify expressions, you must learn what expressions are at their core.
In mathematics, expressions are descriptions that have no less than two terms. These terms can combine numbers, variables, or both and can be connected 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 consist of both numbers (8 and 2) and variables (x and y).
Expressions that include coefficients, variables, and occasionally constants, are also referred to as polynomials.
Simplifying expressions is essential because it opens up the possibility of learning how to solve them. Expressions can be expressed in convoluted ways, and without simplification, you will have a hard time trying to solve them, with more opportunity for a mistake.
Of course, every expression vary in how they're simplified depending on what terms they include, but there are typical steps that are applicable to all rational expressions of real numbers, whether they are square roots, logarithms, or otherwise.
These steps are called the PEMDAS rule, short for parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule shows us the order of operations for expressions.
Parentheses. Solve equations inside the parentheses first by using addition or applying subtraction. If there are terms right outside the parentheses, use the distributive property to apply multiplication the term outside with the one inside.
Exponents. Where possible, use the exponent principles to simplify the terms that have exponents.
Multiplication and Division. If the equation requires it, use multiplication and division to simplify like terms that are applicable.
Addition and subtraction. Lastly, add or subtract the simplified terms in the equation.
Rewrite. Make sure that there are no more like terms that require simplification, then rewrite the simplified equation.
The Rules For Simplifying Algebraic Expressions
In addition to the PEMDAS sequence, there are a few additional principles you need to be aware of when working with algebraic expressions.
You can only apply simplification to terms with common variables. When adding 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 keeping the x as it is.
Parentheses that contain another expression on the 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, as shown here: a(b+c) = ab + ac.
An extension of the distributive property is called the property of multiplication. When two distinct expressions within parentheses are multiplied, the distribution rule applies, and every separate term will will require multiplication by the other terms, making 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 must also need to be distributed, changing the signs of the terms on the inside of the parentheses. Like in this example: -(8x + 2) will turn into -8x - 2.
Likewise, a plus sign outside the parentheses denotes that it will be distributed to the terms inside. However, this means that you are able to 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 principles were simple enough to follow as they only applied to properties that impact simple terms with variables and numbers. Still, there are more rules that you need to apply when working with exponents and expressions.
Here, we will review the principles of exponents. Eight properties affect how we utilize exponentials, those are the following:
Zero Exponent Rule. This property states that any term with a 0 exponent is equal to 1. Or a0 = 1.
Identity Exponent Rule. Any term with a 1 exponent will not alter the value. Or a1 = a.
Product Rule. When two terms with the same variables are multiplied, 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 will subtract their two respective exponents. This is seen as 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 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 being the product of the two exponents applied to it, or (am)n = amn.
Power of a Product Rule. An exponent applied to two terms that have unique variables needs to 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 assume 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 on the inside of a parentheses must be multiplied by all of the expressions inside. Let’s see 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 like with exponents, expressions with fractions also have some rules that you must follow.
When an expression contains fractions, here is what to remember.
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 denominators and numerators.
Simplification. Only fractions at their lowest state should be written in the expression. Apply the PEMDAS property and ensure 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 binomials, decimals, square roots, quadratic equations, logarithms, or linear equations.
Practice Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
Here, the properties that should be noted first are the PEMDAS and the distributive property. The distributive property will distribute 4 to all the expressions inside the parentheses, while PEMDAS will govern the order of simplification.
Due to the distributive property, the term outside 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 the terms with the same variables, and every 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 expresses that the the order should start with expressions within parentheses, and in this case, that expression also necessitates the distributive property. In this scenario, the term y/4 must be distributed amongst 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 set aside the first term for now and simplify the terms with factors assigned to them. Remember we know from PEMDAS that fractions will need to multiply their denominators and numerators individually, 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 utilized to distribute every 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 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
Since there are no more like terms to simplify, this becomes our final answer.
Simplifying Expressions FAQs
What should I remember when simplifying expressions?
When simplifying algebraic expressions, remember that you have to obey the exponential rule, the distributive property, and PEMDAS 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 lowest form.
What is the difference between solving an equation and simplifying an expression?
Solving and simplifying expressions are very different, although, they can be incorporated into the same process the same process because you first need to simplify expressions before you solve them.
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