Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions are one of the most intimidating for budding pupils in their early years of high school or college.
Nevertheless, understanding how to deal with these equations is important because it is primary knowledge that will help them navigate higher arithmetics and complicated problems across various industries.
This article will go over everything you must have to learn simplifying expressions. We’ll cover the laws of simplifying expressions and then validate what we've learned via some sample problems.
How Does Simplifying Expressions Work?
Before learning how to simplify expressions, you must understand what expressions are at their core.
In mathematics, expressions are descriptions that have a minimum of two terms. These terms can combine variables, numbers, or both and can be connected through addition or subtraction.
For example, let’s review the following expression.
8x + 2y - 3
This expression contains three terms; 8x, 2y, and 3. The first two contain both numbers (8 and 2) and variables (x and y).
Expressions containing variables, coefficients, and sometimes constants, are also known as polynomials.
Simplifying expressions is essential because it opens up the possibility of understanding how to solve them. Expressions can be expressed in intricate ways, and without simplifying them, anyone will have a hard time trying to solve them, with more opportunity for error.
Obviously, every expression vary concerning how they are simplified based on what terms they incorporate, but there are common steps that are applicable to all rational expressions of real numbers, regardless of 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 declares the order of operations for expressions.
Parentheses. Solve equations inside the parentheses first by using 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 workable, use the exponent rules to simplify the terms that include exponents.
Multiplication and Division. If the equation necessitates it, use multiplication and division to simplify like terms that are applicable.
Addition and subtraction. Finally, add or subtract the resulting terms of the equation.
Rewrite. Ensure that there are no remaining like terms that need to be simplified, and rewrite the simplified equation.
The Properties For Simplifying Algebraic Expressions
Beyond the PEMDAS principle, there are a few additional principles you need to be aware of when working with algebraic expressions.
You can only simplify terms with common variables. When adding these terms, add the coefficient numbers and leave 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 utilize the distributive property. The distributive property allows you to simplify terms 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 referred to as the principle of multiplication. When two separate expressions within parentheses are multiplied, the distributive property applies, and all individual term will have to be multiplied by the other terms, making each set of equations, common factors of each other. For example: (a + b)(c + d) = a(c + d) + b(c + d).
A negative sign directly outside of 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. Like in this example: -(8x + 2) will turn into -8x - 2.
Similarly, a plus sign outside the parentheses denotes that it will have distribution applied to the terms inside. Despite that, 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 prior principles were easy enough to implement as they only dealt with rules that impact simple terms with variables and numbers. However, there are more rules that you must follow when working with expressions with exponents.
Here, we will discuss the laws of exponents. Eight properties influence how we process exponentials, that includes the following:
Zero Exponent Rule. This property states that any term with the exponent of 0 equals 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 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 the same variables are divided by each other, their quotient will subtract their applicable exponents. This is seen as 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 possess different variables needs to 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 acquire the exponent given, (a/b)m = am/bm.
Simplifying Expressions with the Distributive Property
The distributive property is the principle that states that any term multiplied by an expression within parentheses must be multiplied by all of the expressions on the inside. Let’s see 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 can consist of fractions, and just like with exponents, expressions with fractions also have multiple rules that you have to follow.
When an expression contains fractions, here is 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 numerators and denominators.
Laws of exponents. This shows us that fractions will more likely 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 written in the expression. Refer to the PEMDAS property and ensure that no two terms contain the same variables.
These are the same rules that you can apply when simplifying any real numbers, whether they are square roots, binomials, decimals, quadratic equations, logarithms, or linear equations.
Sample 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 PEMDAS and the distributive property. The distributive property will distribute 4 to all other expressions on the inside of the parentheses, while PEMDAS will dictate the order of simplification.
As a result of the distributive property, the term outside 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, remember to add the terms with matching variables, and all term should be in its most simplified 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 example, that expression also requires the distributive property. In this scenario, the term y/4 will need to be distributed amongst the two terms on the inside of the parentheses, as seen here.
1/3x + y/4(5x) + y/4(2)
Here, let’s set aside the first term for now and simplify the terms with factors attached to them. Since we know from PEMDAS that fractions will need to multiply their numerators and denominators 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 used to distribute all terms 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
Because there are no more 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, remember that you must obey the exponential rule, the distributive property, and PEMDAS rules as well as the principle of multiplication of algebraic expressions. Finally, ensure that every term on your expression is in its lowest form.
How are simplifying expressions and solving equations different?
Solving and simplifying expressions are quite different, however, they can be part of the same process the same process since you have to simplify expressions before solving them.
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