Evaluate The Expression.Write Your Answer As A Fraction Or Whole Number Without Exponents.${ 9^{-3} = }$

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Introduction

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In mathematics, exponents are a fundamental concept used to represent repeated multiplication of a number. When dealing with negative exponents, it's essential to understand how to evaluate them correctly. In this article, we will explore the concept of negative exponents and learn how to evaluate the expression $9^{-3}$.

What are Negative Exponents?

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A negative exponent is a mathematical operation that involves raising a number to a power that is less than zero. In other words, it's the reciprocal of a positive exponent. For example, $a^{-n} = \frac{1}{a^n}$. Negative exponents can be challenging to work with, but understanding their properties is crucial for solving mathematical problems.

Evaluating Negative Exponents

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To evaluate a negative exponent, we need to follow a specific rule. The rule states that $a^{-n} = \frac{1}{a^n}$. This means that to evaluate a negative exponent, we need to take the reciprocal of the base number raised to the power of the positive exponent.

Evaluating the Expression $9^{-3}$

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Now that we understand the concept of negative exponents, let's evaluate the expression $9^{-3}$. Using the rule mentioned earlier, we can rewrite the expression as:

$$9^{-3} = \frac{1}{9^3}$$

Simplifying the Expression

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To simplify the expression, we need to calculate the value of $9^3$. $9^3$ is equal to $9 \times 9 \times 9$, which is equal to $729$. Therefore, the expression can be rewritten as:

$$9^{-3} = \frac{1}{729}$$

Conclusion

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In conclusion, evaluating the expression $9^{-3}$ involves understanding the concept of negative exponents and applying the rule to rewrite the expression as a fraction. By simplifying the expression, we can determine that $9^{-3} = \frac{1}{729}$. This demonstrates the importance of understanding negative exponents in mathematics and how they can be used to evaluate complex expressions.

Frequently Asked Questions

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Q: What is a negative exponent?

A: A negative exponent is a mathematical operation that involves raising a number to a power that is less than zero. It's the reciprocal of a positive exponent.

Q: How do I evaluate a negative exponent?

A: To evaluate a negative exponent, you need to take the reciprocal of the base number raised to the power of the positive exponent.

Q: What is the value of $9^{-3}$?

A: The value of $9^{-3}$ is $\frac{1}{729}$.

Final Thoughts

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Evaluating the expression $9^{-3}$ requires a solid understanding of negative exponents and their properties. By applying the rule and simplifying the expression, we can determine the correct value. This article has provided a comprehensive overview of negative exponents and how to evaluate them correctly. With practice and patience, you can become proficient in working with negative exponents and solving complex mathematical problems.

Additional Resources

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For further learning, we recommend the following resources:

• Khan Academy: Negative Exponents
• Mathway: Negative Exponents
• Wolfram Alpha: Negative Exponents

By exploring these resources, you can gain a deeper understanding of negative exponents and how to apply them in various mathematical contexts.

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Introduction

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In our previous article, we explored the concept of negative exponents and learned how to evaluate the expression $9^{-3}$. However, we understand that there may be many more questions and concerns about negative exponents. In this article, we will address some of the most frequently asked questions about evaluating expressions with negative exponents.

Q&A: Evaluating Expressions with Negative Exponents

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Q: What is the rule for evaluating negative exponents?

A: The rule for evaluating negative exponents is $a^{-n} = \frac{1}{a^n}$. This means that to evaluate a negative exponent, you need to take the reciprocal of the base number raised to the power of the positive exponent.

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

A: To simplify an expression with a negative exponent, you need to follow these steps:

1. Rewrite the expression using the rule $a^{-n} = \frac{1}{a^n}$.
2. Simplify the expression by evaluating the positive exponent.
3. Take the reciprocal of the result.

Q: What is the value of $2^{-4}$?

A: To evaluate the expression $2^{-4}$, we need to follow the rule $a^{-n} = \frac{1}{a^n}$. This means that $2^{-4} = \frac{1}{2^4}$. Simplifying the expression, we get $2^{-4} = \frac{1}{16}$.

Q: How do I evaluate an expression with a negative exponent and a variable?

A: To evaluate an expression with a negative exponent and a variable, you need to follow the same steps as before:

1. Rewrite the expression using the rule $a^{-n} = \frac{1}{a^n}$.
2. Simplify the expression by evaluating the positive exponent.
3. Take the reciprocal of the result.

For example, to evaluate the expression $x^{-3}$, we need to follow these steps:

1. Rewrite the expression using the rule $x^{-3} = \frac{1}{x^3}$.
2. Simplify the expression by evaluating the positive exponent.
3. Take the reciprocal of the result.

Q: What is the value of $x^{-3}$?

A: To evaluate the expression $x^{-3}$, we need to follow the steps mentioned earlier. This means that $x^{-3} = \frac{1}{x^3}$.

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

A: Yes, you can simplify an expression with a negative exponent and a fraction. To do this, you need to follow the same steps as before:

1. Rewrite the expression using the rule $a^{-n} = \frac{1}{a^n}$.
2. Simplify the expression by evaluating the positive exponent.
3. Take the reciprocal of the result.

For example, to simplify the expression $\frac{1}{2^{-3}}$, we need to follow these steps:

1. Rewrite the expression using the rule $\frac{1}{2^{-3}} = 2^3$.
2. Simplify the expression by evaluating the positive exponent.
3. Take the reciprocal of the result.

Q: What is the value of $\frac{1}{2^{-3}}$?

A: To simplify the expression $\frac{1}{2^{-3}}$, we need to follow the steps mentioned earlier. This means that $\frac{1}{2^{-3}} = 2^3 = 8$.

Conclusion

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In conclusion, evaluating expressions with negative exponents requires a solid understanding of the rule $a^{-n} = \frac{1}{a^n}$. By following the steps mentioned in this article, you can simplify expressions with negative exponents and variables. Remember to take the reciprocal of the result to get the final answer.

Frequently Asked Questions

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Q: What is the rule for evaluating negative exponents?

A: The rule for evaluating negative exponents is $a^{-n} = \frac{1}{a^n}$.

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

A: To simplify an expression with a negative exponent, you need to follow these steps:

1. Rewrite the expression using the rule $a^{-n} = \frac{1}{a^n}$.
2. Simplify the expression by evaluating the positive exponent.
3. Take the reciprocal of the result.

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

A: Yes, you can simplify an expression with a negative exponent and a fraction.

Final Thoughts

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Evaluating expressions with negative exponents requires practice and patience. By following the steps mentioned in this article, you can become proficient in simplifying expressions with negative exponents and variables. Remember to take the reciprocal of the result to get the final answer.

Additional Resources

---

For further learning, we recommend the following resources:

• Khan Academy: Negative Exponents
• Mathway: Negative Exponents
• Wolfram Alpha: Negative Exponents

By exploring these resources, you can gain a deeper understanding of negative exponents and how to apply them in various mathematical contexts.