The Power $9^2$ Is Equivalent To 81. What Is The Value Of $9^{-2}$?A. − 81 -81 − 81 B. − 9 -9 − 9 C. 1 81 \frac{1}{81} 81 1 D. 1 9 \frac{1}{9} 9 1
Introduction
When it comes to solving equations involving exponents, understanding the concept of negative exponents is crucial. In this article, we will delve into the world of negative exponents and explore how to solve equations that involve them. We will use the example of the equation $9^{-2}$ to illustrate the concept and provide a step-by-step guide on how to solve it.
What are Negative Exponents?
A negative exponent is a mathematical operation that involves raising a number to a power that is less than 1. In other words, it is the reciprocal of a positive exponent. For example, $a^{-n} = \frac{1}{a^n}$. This means that if we have a negative exponent, we can rewrite it as a fraction with 1 as the numerator and the base raised to the power of the absolute value of the exponent as the denominator.
Solving the Equation $9^{-2}$
To solve the equation $9^{-2}$, we need to understand that the negative exponent indicates that we need to take the reciprocal of the base raised to the power of the absolute value of the exponent. In this case, the base is 9 and the exponent is -2. To solve the equation, we can rewrite it as:
Evaluating the Expression
Now that we have rewritten the equation, we can evaluate the expression. We know that $9^2 = 81$, so we can substitute this value into the equation:
Conclusion
In conclusion, the value of $9^{-2}$ is $\frac{1}{81}$. This is because the negative exponent indicates that we need to take the reciprocal of the base raised to the power of the absolute value of the exponent. By rewriting the equation and evaluating the expression, we can see that the correct answer is $\frac{1}{81}$.
Common Mistakes to Avoid
When solving equations involving negative exponents, there are several common mistakes to avoid. These include:
- Not understanding the concept of negative exponents: Negative exponents are the reciprocal of positive exponents. It is essential to understand this concept to solve equations involving negative exponents.
- Not rewriting the equation correctly: When rewriting the equation, it is crucial to ensure that the base and exponent are correct. A small mistake can lead to an incorrect solution.
- Not evaluating the expression correctly: When evaluating the expression, it is essential to ensure that the correct values are substituted into the equation.
Tips and Tricks
When solving equations involving negative exponents, here are some tips and tricks to keep in mind:
- Use the definition of negative exponents: The definition of negative exponents is $a^{-n} = \frac{1}{a^n}$. This definition can be used to rewrite the equation and solve it.
- Rewrite the equation carefully: When rewriting the equation, ensure that the base and exponent are correct. A small mistake can lead to an incorrect solution.
- Evaluate the expression carefully: When evaluating the expression, ensure that the correct values are substituted into the equation.
Real-World Applications
Negative exponents have several real-world applications. These include:
- Science and engineering: Negative exponents are used to describe the behavior of physical systems, such as the decay of radioactive materials.
- Finance: Negative exponents are used to calculate the return on investment (ROI) of a financial instrument.
- Computer science: Negative exponents are used to describe the behavior of algorithms and data structures.
Conclusion
In conclusion, negative exponents are a fundamental concept in mathematics that can be used to solve equations involving exponents. By understanding the concept of negative exponents and following the steps outlined in this article, you can solve equations involving negative exponents with confidence. Remember to use the definition of negative exponents, rewrite the equation carefully, and evaluate the expression carefully to ensure that you get the correct solution.
Final Answer
The final answer is:
Introduction
In our previous article, we explored the concept of negative exponents and how to solve equations involving them. In this article, we will answer some of the most frequently asked questions about negative exponents.
Q: What is the definition of a negative exponent?
A: A negative exponent is a mathematical operation that involves raising a number to a power that is less than 1. In other words, it is the reciprocal of a positive exponent. For example, $a^{-n} = \frac{1}{a^n}$.
Q: How do I rewrite an equation with a negative exponent?
A: To rewrite an equation with a negative exponent, you need to take the reciprocal of the base raised to the power of the absolute value of the exponent. For example, $9^{-2} = \frac{1}{9^2}$.
Q: What is the difference between a negative exponent and a positive exponent?
A: A negative exponent is the reciprocal of a positive exponent. For example, $a^{-n} = \frac{1}{a^n}$. This means that if you have a negative exponent, you can rewrite it as a fraction with 1 as the numerator and the base raised to the power of the absolute value of the exponent as the denominator.
Q: Can I simplify an equation with a negative exponent?
A: Yes, you can simplify an equation with a negative exponent by rewriting it as a fraction with 1 as the numerator and the base raised to the power of the absolute value of the exponent as the denominator. For example, $9^{-2} = \frac{1}{9^2}$.
Q: How do I evaluate an expression with a negative exponent?
A: To evaluate an expression with a negative exponent, you need to substitute the correct values into the equation. For example, if you have the equation $9^{-2} = \frac{1}{9^2}$, you can substitute $9^2 = 81$ into the equation to get $\frac{1}{81}$.
Q: Can I use negative exponents in real-world applications?
A: Yes, negative exponents have several real-world applications. These include science and engineering, finance, and computer science.
Q: What are some common mistakes to avoid when working with negative exponents?
A: Some common mistakes to avoid when working with negative exponents include not understanding the concept of negative exponents, not rewriting the equation correctly, and not evaluating the expression correctly.
Q: How can I practice working with negative exponents?
A: You can practice working with negative exponents by solving equations and expressions that involve negative exponents. You can also try using online resources or math software to help you practice.
Q: What are some tips and tricks for working with negative exponents?
A: Some tips and tricks for working with negative exponents include using the definition of negative exponents, rewriting the equation carefully, and evaluating the expression carefully.
Conclusion
In conclusion, negative exponents are a fundamental concept in mathematics that can be used to solve equations involving exponents. By understanding the concept of negative exponents and following the steps outlined in this article, you can solve equations involving negative exponents with confidence. Remember to use the definition of negative exponents, rewrite the equation carefully, and evaluate the expression carefully to ensure that you get the correct solution.
Final Answer
The final answer is:
Additional Resources
If you want to learn more about negative exponents, here are some additional resources that you can use:
- Math textbooks: You can find math textbooks that cover the topic of negative exponents in your local library or online.
- Online resources: There are many online resources that can help you learn about negative exponents, including video tutorials, interactive lessons, and practice problems.
- Math software: You can use math software such as Mathematica or Maple to help you practice working with negative exponents.
- Online communities: You can join online communities such as Reddit's r/learnmath or r/math to ask questions and get help from other math enthusiasts.