Problem 122 - Elementary Algebra

Question

Simplify the following problems. $$ \frac{18 x^{4 n+9}}{2 x^{2 n+1}} $$

Step-by-Step Solution

Answer

Question: Simplify the expression $\frac{18x^{4n+9}}{2x^{2n+1}}$. Answer: $9x^{2n+8}$

Step 1: Simplify the constants

To simplify the constants, we just need to divide them. So, we have: $$ \frac{18}{2} = 9 $$

Step 2: Simplify the variable part

To simplify the variable part, we first need to recognize that both exponents have the same base, x. According to the rules of exponents, when two exponentials with the same base are divided, we can subtract the exponents: $$ \frac{x^{4n+9}}{x^{2n+1}} = x^{(4n+9)-(2n+1)} $$ Now, we can distribute the subtraction to the terms inside the parentheses: $$ x^{(4n+9)-(2n+1)} = x^{(4n-2n)+(9-1)} = x^{2n+8} $$

Step 3: Combine constants and variable parts

Now that we have simplified the constants (from Step 1) and the variable part (from Step 2), we just need to combine them together to get the final simplified expression: $$ 9x^{2n+8} $$

Key Concepts

Simplifying Constants

When dealing with algebraic expressions, one of the initial steps is often to simplify the constants involved. Constants are numbers that have a fixed value, unlike variables whose value can change. In the given exercise, the simplification process starts with the numerical part of the fraction. By dividing 18 by 2, we get 9.

This operation may seem straightforward, but it is essential for reducing the complexity of the problem before dealing with more complicated variables and exponents. It's important always to perform all possible simplifications of constants at the beginning. This includes looking out for opportunities to add, subtract, multiply, divide, or factor out constants to make the expression as simple as possible before proceeding to variable terms.

Rules of Exponents

Understanding the rules of exponents is crucial when simplifying algebraic expressions with variables raised to powers. One such rule states that when you divide exponential expressions that have the same base, you can subtract the exponents.

In the given expression, \( \frac{x^{4n+9}}{x^{2n+1}} \), we have two powers of x. By applying the rule, we can simplify the expression to \( x^{(4n+9)-(2n+1)} \), which further simplifies to \( x^{2n+8} \). Remember that this rule only works when the bases are identical. Always pay close attention to the base of your exponents, as this will determine which exponent rule to apply.

Variable Expression Simplification

After simplifying constants and applying rules of exponents, the next step is to simplify the variable expression itself. Variable expressions may include letters (variables) that stand in for unknown values, combined with known values (constants). For instance, in our exercise, after applying the rules of exponents, we're left with \( x^{2n+8} \).

This result is already in its simplest form, as it's not possible to combine the terms further since they are not like terms. Simplifying variable expressions can involve a number of techniques such as combining like terms, factoring, expanding or using exponent rules. The goal is to write the expression in the most concise and clear way possible.