Select All The Expressions That Are Equivalent To $\frac{9^{-20}}{9^4}$.A. $9^{-5}$ B. $\frac{1}{9^{24}}$ C. $9^{-24}$ D. $9^{-16}$

Introduction

Exponential expressions are a fundamental concept in mathematics, and understanding how to simplify them is crucial for solving various mathematical problems. In this article, we will focus on simplifying the expression $\frac{9^{-20}}{9^4}$ and select the equivalent expressions from the given options.

Understanding Exponential Notation

Before we dive into simplifying the expression, let's briefly review exponential notation. When we write $a^b$, it means $a$ raised to the power of $b$. For example, $2^3$ means $2$ multiplied by itself $3$ times, which equals $8$. Similarly, $9^{-20}$ means $9$ multiplied by itself $-20$ times.

Simplifying the Expression

To simplify the expression $\frac{9^{-20}}{9^4}$, we can use the quotient rule of exponents, which states that when we divide two exponential expressions with the same base, we subtract the exponents. In this case, the base is $9$, and the exponents are $-20$ and $4$.

Using the quotient rule, we can rewrite the expression as:

$$\frac{9^{-20}}{9^4} = 9^{-20-4} = 9^{-24}$$

Selecting Equivalent Expressions

Now that we have simplified the expression, let's select the equivalent expressions from the given options.

• Option A: $9^{-5}$ - This expression is not equivalent to $\frac{9^{-20}}{9^4}$ because the exponents are different.
• Option B: $\frac{1}{9^{24}}$ - This expression is equivalent to $\frac{9^{-20}}{9^4}$ because we can rewrite it as $9^{-24}$.
• Option C: $9^{-24}$ - This expression is equivalent to $\frac{9^{-20}}{9^4}$ because we simplified the expression to $9^{-24}$.
• Option D: $9^{-16}$ - This expression is not equivalent to $\frac{9^{-20}}{9^4}$ because the exponents are different.

Conclusion

In conclusion, the equivalent expressions to $\frac{9^{-20}}{9^4}$ are $9^{-24}$ and $\frac{1}{9^{24}}$. We simplified the expression using the quotient rule of exponents and selected the equivalent expressions from the given options.

Final Answer

The final answer is:

• Option B: $\frac{1}{9^{24}}$
• Option C: $9^{-24}$

Q&A: Simplifying Exponential Expressions

Q: What is the quotient rule of exponents?

A: The quotient rule of exponents states that when we divide two exponential expressions with the same base, we subtract the exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$.

Q: How do I simplify the expression $\frac{9^{-20}}{9^4}$?

A: To simplify the expression $\frac{9^{-20}}{9^4}$, we can use the quotient rule of exponents. We subtract the exponents, which gives us $9^{-20-4} = 9^{-24}$.

Q: What is the equivalent expression to $\frac{9^{-20}}{9^4}$?

A: The equivalent expressions to $\frac{9^{-20}}{9^4}$ are $9^{-24}$ and $\frac{1}{9^{24}}$.

Q: How do I select the equivalent expressions from the given options?

A: To select the equivalent expressions, we need to compare the simplified expression with the given options. We can rewrite the expression $\frac{1}{9^{24}}$ as $9^{-24}$, which is equivalent to the simplified expression.

Q: What are the options for equivalent expressions?

A: The options for equivalent expressions are:

• Option A: $9^{-5}$ - This expression is not equivalent to $\frac{9^{-20}}{9^4}$ because the exponents are different.
• Option B: $\frac{1}{9^{24}}$ - This expression is equivalent to $\frac{9^{-20}}{9^4}$ because we can rewrite it as $9^{-24}$.
• Option C: $9^{-24}$ - This expression is equivalent to $\frac{9^{-20}}{9^4}$ because we simplified the expression to $9^{-24}$.
• Option D: $9^{-16}$ - This expression is not equivalent to $\frac{9^{-20}}{9^4}$ because the exponents are different.

Q: What is the final answer?

A: The final answer is:

• Option B: $\frac{1}{9^{24}}$
• Option C: $9^{-24}$

Common Mistakes to Avoid

• Not using the quotient rule of exponents: When dividing two exponential expressions with the same base, we need to subtract the exponents.
• Not rewriting the expression: We need to rewrite the expression $\frac{1}{9^{24}}$ as $9^{-24}$ to compare it with the simplified expression.
• Not selecting the correct options: We need to carefully compare the simplified expression with the given options to select the correct equivalent expressions.

Conclusion

In conclusion, simplifying exponential expressions requires a clear understanding of the quotient rule of exponents and the ability to rewrite expressions. By following the steps outlined in this article, you can simplify complex expressions and select the correct equivalent expressions.