Determine The Following Without The Use Of A Calculator: $9^{-\frac{3}{2}}$
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Introduction
In mathematics, exponents are a fundamental concept that can be used to represent large numbers in a more manageable way. However, when dealing with negative exponents, it can be challenging to determine their values without the use of a calculator. In this article, we will explore how to determine the value of the expression $9^{-\frac{3}{2}}$ without the use of a calculator.
Understanding Negative Exponents
Before we dive into the solution, it's essential to understand what negative exponents represent. A negative exponent is a shorthand way of writing a fraction with a negative exponent. For example, $a^{-n}$ is equivalent to $\frac{1}{a^n}$. This means that when we see a negative exponent, we can rewrite it as a fraction with a positive exponent.
Rewriting the Expression
Using the definition of negative exponents, we can rewrite the expression $9^{-\frac{3}{2}}$ as $\frac{1}{9^{\frac{3}{2}}}$.
Simplifying the Expression
Now that we have rewritten the expression, we can simplify it further. To do this, we need to understand the properties of exponents. Specifically, we need to recall that when we raise a power to a power, we multiply the exponents. In this case, we have $9^{\frac{3}{2}}$, which can be rewritten as $(9{\frac{1}{2}})3$.
Understanding Square Roots
Before we can simplify the expression further, we need to understand what square roots represent. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 multiplied by 4 equals 16. In this case, we have $9^{\frac{1}{2}}$, which represents the square root of 9.
Simplifying the Square Root
Now that we have understood what square roots represent, we can simplify the expression further. The square root of 9 is 3, because 3 multiplied by 3 equals 9. Therefore, we can rewrite $9^{\frac{1}{2}}$ as 3.
Simplifying the Expression Further
Now that we have simplified the square root, we can simplify the expression further. We have $(9{\frac{1}{2}})3$, which can be rewritten as $3^3$.
Evaluating the Expression
Finally, we can evaluate the expression $3^3$. This means multiplying 3 by itself three times: $3 \times 3 \times 3 = 27$.
Conclusion
In conclusion, we have determined the value of the expression $9^{-\frac{3}{2}}$ without the use of a calculator. By rewriting the expression as a fraction with a positive exponent, simplifying the square root, and evaluating the expression, we have arrived at the final answer of $\frac{1}{27}$.
Frequently Asked Questions
Q: What is the definition of a negative exponent?
A: A negative exponent is a shorthand way of writing a fraction with a negative exponent. For example, $a^{-n}$ is equivalent to $\frac{1}{a^n}$.
Q: How do I simplify a negative exponent?
A: To simplify a negative exponent, you can rewrite it as a fraction with a positive exponent. For example, $9^{-\frac{3}{2}}$ can be rewritten as $\frac{1}{9^{\frac{3}{2}}}$.
Q: What is the property of exponents that allows me to simplify the expression?
A: The property of exponents that allows you to simplify the expression is the rule that when you raise a power to a power, you multiply the exponents. In this case, we have $(9{\frac{1}{2}})3$, which can be rewritten as $9^{\frac{3}{2}}$.
Q: What is the final answer to the expression $9^{-\frac{3}{2}}$?
A: The final answer to the expression $9^{-\frac{3}{2}}$ is $\frac{1}{27}$.
Additional Resources
For more information on exponents and how to simplify them, check out the following resources:
- Khan Academy: Exponents and Exponential Functions
- Mathway: Exponents and Exponential Functions
- Wolfram Alpha: Exponents and Exponential Functions
Final Thoughts
In conclusion, determining the value of the expression $9^{-\frac{3}{2}}$ without the use of a calculator requires a solid understanding of exponents and how to simplify them. By rewriting the expression as a fraction with a positive exponent, simplifying the square root, and evaluating the expression, we have arrived at the final answer of $\frac{1}{27}$. We hope this article has provided you with a better understanding of how to determine exponents without a calculator.
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Introduction
In our previous article, we explored how to determine the value of the expression $9^{-\frac{3}{2}}$ without the use of a calculator. We broke down the problem step by step, using the properties of exponents to simplify the expression and arrive at the final answer of $\frac{1}{27}$. In this article, we will continue to explore the topic of exponents and answer some frequently asked questions.
Q&A: Determining Exponents without a Calculator
Q: What is the definition of a negative exponent?
A: A negative exponent is a shorthand way of writing a fraction with a negative exponent. For example, $a^{-n}$ is equivalent to $\frac{1}{a^n}$.
Q: How do I simplify a negative exponent?
A: To simplify a negative exponent, you can rewrite it as a fraction with a positive exponent. For example, $9^{-\frac{3}{2}}$ can be rewritten as $\frac{1}{9^{\frac{3}{2}}}$.
Q: What is the property of exponents that allows me to simplify the expression?
A: The property of exponents that allows you to simplify the expression is the rule that when you raise a power to a power, you multiply the exponents. In this case, we have $(9{\frac{1}{2}})3$, which can be rewritten as $9^{\frac{3}{2}}$.
Q: What is the final answer to the expression $9^{-\frac{3}{2}}$?
A: The final answer to the expression $9^{-\frac{3}{2}}$ is $\frac{1}{27}$.
Q: How do I determine the value of an expression with a negative exponent?
A: To determine the value of an expression with a negative exponent, you can follow these steps:
- Rewrite the negative exponent as a fraction with a positive exponent.
- Simplify the expression using the properties of exponents.
- Evaluate the expression to arrive at the final answer.
Q: What are some common mistakes to avoid when simplifying negative exponents?
A: Some common mistakes to avoid when simplifying negative exponents include:
- Not rewriting the negative exponent as a fraction with a positive exponent.
- Not simplifying the expression using the properties of exponents.
- Not evaluating the expression to arrive at the final answer.
Q: Can I use a calculator to determine the value of an expression with a negative exponent?
A: While it is possible to use a calculator to determine the value of an expression with a negative exponent, it is not necessary. By following the steps outlined above, you can determine the value of the expression without the use of a calculator.
Additional Resources
For more information on exponents and how to simplify them, check out the following resources:
- Khan Academy: Exponents and Exponential Functions
- Mathway: Exponents and Exponential Functions
- Wolfram Alpha: Exponents and Exponential Functions
Final Thoughts
In conclusion, determining the value of an expression with a negative exponent requires a solid understanding of exponents and how to simplify them. By following the steps outlined above and avoiding common mistakes, you can determine the value of the expression without the use of a calculator. We hope this article has provided you with a better understanding of how to determine exponents without a calculator.
Common Exponent Problems
Problem 1: $2^{-3}$
A: $\frac{1}{2^3} = \frac{1}{8}$
Problem 2: $5^{-2}$
A: $\frac{1}{5^2} = \frac{1}{25}$
Problem 3: $3^{-4}$
A: $\frac{1}{3^4} = \frac{1}{81}$
Practice Exercises
Exercise 1: Determine the value of $4^{-2}$ without the use of a calculator.
A: $\frac{1}{4^2} = \frac{1}{16}$
Exercise 2: Determine the value of $6^{-3}$ without the use of a calculator.
A: $\frac{1}{6^3} = \frac{1}{216}$
Exercise 3: Determine the value of $8^{-1}$ without the use of a calculator.
A: $\frac{1}{8^1} = \frac{1}{8}$
Conclusion
In conclusion, determining the value of an expression with a negative exponent requires a solid understanding of exponents and how to simplify them. By following the steps outlined above and avoiding common mistakes, you can determine the value of the expression without the use of a calculator. We hope this article has provided you with a better understanding of how to determine exponents without a calculator.