Simplify Exponential Expressions: A Step-by-Step Guide

Let's dive into simplifying the expression $\frac{\left(x^{-7 / 6} vert x^3\right)^{-2 / 3}}{x^{5 / 3}}$. This involves several steps using the properties of exponents. Understanding these properties is key to making the simplification process straightforward and manageable. We will break down each step, ensuring clarity and a thorough understanding of the underlying principles. First, we'll address the terms inside the parentheses, then we'll handle the outer exponent, and finally, we'll deal with the division. This methodical approach will help avoid confusion and ensure accuracy. Remember, the goal is to combine like terms and reduce the expression to its simplest form, making it easier to work with in further calculations or analyses. By the end of this guide, you'll be well-equipped to tackle similar simplification problems with confidence and ease.

Understanding the Basics of Exponents

Before we begin, it's crucial to understand the fundamental properties of exponents. These properties are the building blocks for simplifying complex expressions. Let's briefly review some essential rules:

1. Product of Powers: $x^a vert x^b = x^{a+b}$ (When multiplying like bases, add the exponents).
2. Power of a Power: $(x^a)^b = x^{ab}$ (When raising a power to a power, multiply the exponents).
3. Quotient of Powers: $\frac{x^a}{x^b} = x^{a-b}$ (When dividing like bases, subtract the exponents).
4. Negative Exponent: $x^{-a} = \frac{1}{x^a}$ (A negative exponent indicates a reciprocal).

These rules are essential for manipulating and simplifying exponential expressions. Mastering them will enable you to tackle more complex problems with greater confidence. Throughout this guide, we will apply these rules step-by-step to simplify the given expression. Keep these rules in mind as we proceed, and you'll find the process much more intuitive and manageable. Understanding these basics is not just about memorization; it's about recognizing how and when to apply each rule to achieve the desired simplification.

Step-by-Step Simplification

Step 1: Simplify Inside the Parentheses

Our first task is to simplify the expression inside the parentheses: $x^{-7 / 6} vert x^3$. According to the product of powers rule, we add the exponents:

$x^{-7 / 6} vert x^3 = x^{(-7 / 6) + 3}$.

To add the exponents, we need a common denominator. Convert 3 to a fraction with a denominator of 6: $3 = \frac{18}{6}$.

So, the expression becomes:

$x^{(-7 / 6) + (18 / 6)} = x^{(18 - 7) / 6} = x^{11 / 6}$.

Now we have simplified the inside of the parentheses to $x^{11 / 6}$. This simplification is crucial because it combines the two terms into a single term with a single exponent. This makes the subsequent steps much easier to handle. Remember, the goal is to reduce the complexity of the expression at each step, making it more manageable. By simplifying the inside of the parentheses first, we've set the stage for further simplification using the other properties of exponents.

Step 2: Apply the Outer Exponent

Next, we need to apply the outer exponent, which is $-\frac{2}{3}$, to the simplified term $x^{11 / 6}$. Using the power of a power rule, we multiply the exponents:

$(x^{11 / 6})^{-2 / 3} = x^{(11 / 6) vert (-2 / 3)}$.

Multiplying the fractions, we get:

$\frac{11}{6} vert \frac{-2}{3} = \frac{11 vert -2}{6 vert 3} = \frac{-22}{18}$.

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

$\frac{-22}{18} = \frac{-11}{9}$.

Therefore, $(x^{11 / 6})^{-2 / 3} = x^{-11 / 9}$. Applying the outer exponent has now given us a single term with a single exponent. This exponent, however, is negative, which we will address in a later step if needed. For now, we have successfully handled the power of a power and simplified the expression further. Remember to always simplify fractions to their lowest terms to keep the expression as clean and manageable as possible.

Step 3: Divide by $x^{5 / 3}$

Now we have to divide our simplified expression, $x^{-11 / 9}$, by $x^{5 / 3}$. Using the quotient of powers rule, we subtract the exponents:

$\frac{x^{-11 / 9}}{x^{5 / 3}} = x^{(-11 / 9) - (5 / 3)}$.

To subtract the exponents, we need a common denominator. Convert $\frac{5}{3}$ to a fraction with a denominator of 9: $\frac{5}{3} = \frac{15}{9}$.

So, the expression becomes:

$x^{(-11 / 9) - (15 / 9)} = x^{(-11 - 15) / 9} = x^{-26 / 9}$.

Now we have combined the terms from the numerator and the denominator into a single term with a single exponent. This simplification is another critical step in reducing the complexity of the original expression. By applying the quotient of powers rule, we've further consolidated the terms, making it easier to understand the final form of the expression.

Step 4: Express with a Positive Exponent (Optional)

While $x^{-26 / 9}$ is a simplified form, we can express it with a positive exponent using the negative exponent rule:

$x^{-26 / 9} = \frac{1}{x^{26 / 9}}$.

This step is optional, and the choice to express the answer with a positive exponent often depends on the context or specific instructions. Both forms, $x^{-26 / 9}$ and $\frac{1}{x^{26 / 9}}$, are mathematically equivalent and represent the simplified expression. If the context requires a positive exponent, then this final transformation is necessary.

Final Simplified Expression

Therefore, the simplified expression is:

$\frac{1}{x^{26 / 9}}$ or $x^{-26 / 9}$.

Conclusion

In summary, we have successfully simplified the given expression $\frac{\left(x^{-7 / 6} vert x^3\right)^{-2 / 3}}{x^{5 / 3}}$ by applying the properties of exponents step-by-step. We first simplified the expression inside the parentheses, then applied the outer exponent, and finally, performed the division. The resulting expression is $\frac{1}{x^{26 / 9}}$ or $x^{-26 / 9}$. Each step involved a clear application of exponent rules, making the process manageable and easy to follow. Understanding and applying these rules is essential for simplifying more complex algebraic expressions. Remember to always look for opportunities to combine like terms and reduce the expression to its simplest form.

By breaking down the problem into smaller, manageable steps, we were able to simplify the expression effectively. This approach not only makes the process less daunting but also reduces the likelihood of errors. Always double-check your work and ensure that you have applied the correct rules at each step. With practice, you'll become more confident and proficient in simplifying exponential expressions.

For further learning on the properties of exponents, you can check out resources like Khan Academy's Exponent Rules.