Simplify The Given Expression. Assume $x \neq 0$. $\left(\frac{20 X^{-3}}{10 X^{-1}}\right)^{-2}$A. $\frac{x^4}{2}$ B. $\frac{x^4}{4}$ C. $\frac{x^2}{4}$

Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently and accurately. Exponential expressions, in particular, can be challenging to simplify, but with the right techniques and strategies, we can master them. In this article, we will focus on simplifying the given expression $\left(\frac{20 x^{-3}}{10 x^{-1}}\right)^{-2}$, assuming $x \neq 0$. We will break down the solution into manageable steps, using the properties of exponents and algebraic manipulations.

Understanding Exponents

Before we dive into the solution, let's review the basics of exponents. An exponent is a small number that is raised to a power, indicating how many times the base is multiplied by itself. For example, $x^2$ means $x$ multiplied by itself twice, or $x \cdot x$. When we have a negative exponent, it means we are taking the reciprocal of the base raised to the positive exponent. For instance, $x^{-2}$ is equivalent to $\frac{1}{x^2}$.

Step 1: Simplify the Fraction

The given expression is $\left(\frac{20 x^{-3}}{10 x^{-1}}\right)^{-2}$. To simplify this expression, we need to start by simplifying the fraction inside the parentheses. We can do this by canceling out common factors in the numerator and denominator.

\frac{20 x^{-3}}{10 x^{-1}} = \frac{2 x^{-3}}{x^{-1}} = 2 x^{-3 - (-1)} = 2 x^{-2}

Step 2: Apply the Negative Exponent

Now that we have simplified the fraction, we can apply the negative exponent outside the parentheses. When we have a negative exponent, we take the reciprocal of the base raised to the positive exponent. In this case, we have $\left(2 x^{-2}\right)^{-2}$.

\left(2 x^{-2}\right)^{-2} = \frac{1}{\left(2 x^{-2}\right)^2} = \frac{1}{4 x^{-4}}

Step 3: Simplify the Expression

Now that we have applied the negative exponent, we can simplify the expression further. We can do this by canceling out common factors in the numerator and denominator.

\frac{1}{4 x^{-4}} = \frac{x^4}{4}

Conclusion

In conclusion, we have successfully simplified the given expression $\left(\frac{20 x^{-3}}{10 x^{-1}}\right)^{-2}$, assuming $x \neq 0$. We broke down the solution into manageable steps, using the properties of exponents and algebraic manipulations. By following these steps, we can simplify even the most complex exponential expressions.

Final Answer

The final answer is $\boxed{\frac{x^4}{4}}$.

Discussion

This problem requires a deep understanding of exponents and algebraic manipulations. The key to solving this problem is to simplify the fraction inside the parentheses and then apply the negative exponent outside the parentheses. By following these steps, we can simplify even the most complex exponential expressions.

Common Mistakes

When simplifying exponential expressions, it's easy to make mistakes. Here are some common mistakes to avoid:

• Not simplifying the fraction inside the parentheses
• Not applying the negative exponent correctly
• Not canceling out common factors in the numerator and denominator

By avoiding these common mistakes, we can ensure that our solutions are accurate and efficient.

Real-World Applications

Simplifying exponential expressions has many real-world applications. For example, in physics, we use exponential expressions to describe the behavior of particles and waves. In engineering, we use exponential expressions to model population growth and decay. By mastering the art of simplifying exponential expressions, we can solve complex problems in a variety of fields.

Conclusion

Introduction

In our previous article, we explored the process of simplifying exponential expressions. We broke down the solution into manageable steps, using the properties of exponents and algebraic manipulations. In this article, we will continue to build on that knowledge by answering some of the most frequently asked questions about simplifying exponential expressions.

Q: What is the order of operations when simplifying exponential expressions?

A: When simplifying exponential expressions, the order of operations is crucial. We need to follow the order of operations (PEMDAS):

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify an expression with multiple negative exponents?

A: When simplifying an expression with multiple negative exponents, we need to apply the rule that states:

$$a^{-m} = \frac{1}{a^m}$$

For example, if we have the expression $\left(\frac{2 x^{-3}}{3 x^{-2}}\right)^{-2}$, we can simplify it as follows:

\left(\frac{2 x^{-3}}{3 x^{-2}}\right)^{-2} = \left(\frac{3 x^{-2}}{2 x^{-3}}\right)^2 = \left(\frac{3}{2} x^{-2 + 3}\right)^2 = \left(\frac{3}{2} x^1\right)^2 = \frac{9}{4} x^2

Q: How do I simplify an expression with a negative exponent and a fraction?

A: When simplifying an expression with a negative exponent and a fraction, we need to apply the rule that states:

$$\frac{1}{a^{-m}} = a^m$$

For example, if we have the expression $\frac{1}{x^{-2}}$, we can simplify it as follows:

\frac{1}{x^{-2}} = x^2

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent indicates that we are multiplying the base by itself a certain number of times. For example, $x^2$ means $x$ multiplied by itself twice, or $x \cdot x$.

A negative exponent, on the other hand, indicates that we are taking the reciprocal of the base raised to the positive exponent. For example, $x^{-2}$ is equivalent to $\frac{1}{x^2}$.

Q: How do I simplify an expression with a zero exponent?

A: When simplifying an expression with a zero exponent, we need to apply the rule that states:

$$a^0 = 1$$

For example, if we have the expression $x^0$, we can simplify it as follows:

x^0 = 1

Q: What are some common mistakes to avoid when simplifying exponential expressions?

A: Some common mistakes to avoid when simplifying exponential expressions include:

• Not simplifying the fraction inside the parentheses
• Not applying the negative exponent correctly
• Not canceling out common factors in the numerator and denominator
• Not following the order of operations (PEMDAS)

By avoiding these common mistakes, we can ensure that our solutions are accurate and efficient.

Conclusion

In conclusion, simplifying exponential expressions is a crucial skill that requires a deep understanding of exponents and algebraic manipulations. By following the steps outlined in this article and avoiding common mistakes, we can simplify even the most complex exponential expressions. Whether we are solving problems in mathematics, physics, or engineering, mastering the art of simplifying exponential expressions is essential for success.