Which Expression Is Equivalent To $9^{-2}$?A. $-81$ B. $ − 18 -18 − 18 [/tex] C. $\frac{1}{81}$ D. $\frac{1}{18}$
Introduction
Exponents are a fundamental concept in mathematics, used to represent repeated multiplication of a number. In this article, we will explore the concept of exponents and how to find equivalent expressions. We will focus on the expression $9^{-2}$ and determine which of the given options is equivalent to it.
What are Exponents?
Exponents are a shorthand way of representing repeated multiplication of a number. For example, $2^3$ represents $2 \times 2 \times 2$, which equals $8$. Exponents are written as a small number raised to a power, indicating how many times the base number should be multiplied by itself.
Negative Exponents
A negative exponent is a fraction with a positive exponent in the denominator. For example, $2^{-3}$ represents $\frac{1}{2^3}$, which equals $\frac{1}{8}$. Negative exponents can be rewritten as positive exponents by flipping the fraction. For example, $2^{-3} = \frac{1}{2^3}$.
Equivalent Expressions
Equivalent expressions are expressions that have the same value, but may be written differently. For example, $2 \times 3$ and $6$ are equivalent expressions, as they both equal $6$. In the context of exponents, equivalent expressions can be found by applying the rules of exponents.
Rules of Exponents
There are several rules of exponents that can be used to find equivalent expressions. These rules include:
- Product of Powers Rule: $a^m \times a^n = a^{m+n}$
- Power of a Power Rule: $(am)n = a^{m \times n}$
- Quotient of Powers Rule: $\frac{am}{an} = a^{m-n}$
Applying the Rules of Exponents
To find the equivalent expression for $9^{-2}$, we can apply the rules of exponents. Using the quotient of powers rule, we can rewrite $9^{-2}$ as $\frac{1}{9^2}$.
Simplifying the Expression
Now that we have rewritten $9^{-2}$ as $\frac{1}{9^2}$, we can simplify the expression. Using the power of a power rule, we can rewrite $9^2$ as $(32)2$, which equals $3^4$.
Finding the Equivalent Expression
Now that we have simplified the expression, we can find the equivalent expression. Using the quotient of powers rule, we can rewrite $\frac{1}{9^2}$ as $\frac{1}{3^4}$.
Conclusion
In conclusion, the equivalent expression for $9^{-2}$ is $\frac{1}{81}$. This can be found by applying the rules of exponents and simplifying the expression.
Answer
The correct answer is C. $\frac{1}{81}$.
Discussion
This problem requires the application of the rules of exponents and simplification of expressions. It is an important concept in mathematics, as it allows us to rewrite expressions in different forms and solve problems more easily.
Related Topics
- Exponents and powers
- Rules of exponents
- Simplifying expressions
- Equivalent expressions
Practice Problems
- Find the equivalent expression for $2^{-3}$.
- Simplify the expression $\frac{1}{4^2}$.
- Find the equivalent expression for $5^{-2}$.
References
- [1] "Exponents and Powers" by Math Open Reference
- [2] "Rules of Exponents" by Khan Academy
- [3] "Simplifying Expressions" by Purplemath
Q&A: Exponents and Equivalent Expressions =============================================
Frequently Asked Questions
Q: What is the difference between a positive exponent and a negative exponent? A: A positive exponent represents repeated multiplication of a number, while a negative exponent represents a fraction with a positive exponent in the denominator.
Q: How do I rewrite a negative exponent as a positive exponent? A: To rewrite a negative exponent as a positive exponent, flip the fraction. For example, $2^{-3} = \frac{1}{2^3}$.
Q: What is the product of powers rule? A: The product of powers rule states that $a^m \times a^n = a^{m+n}$.
Q: What is the power of a power rule? A: The power of a power rule states that $(am)n = a^{m \times n}$.
Q: What is the quotient of powers rule? A: The quotient of powers rule states that $\frac{am}{an} = a^{m-n}$.
Q: How do I apply the rules of exponents to find an equivalent expression? A: To apply the rules of exponents, identify the type of exponent (positive or negative) and the operation (multiplication or division). Then, use the corresponding rule to rewrite the expression.
Q: Can I simplify an expression with a negative exponent? A: Yes, you can simplify an expression with a negative exponent by applying the rules of exponents. For example, $\frac{1}{9^2}$ can be simplified to $\frac{1}{3^4}$.
Q: How do I find the equivalent expression for a given exponent? A: To find the equivalent expression for a given exponent, apply the rules of exponents and simplify the expression. For example, to find the equivalent expression for $9^{-2}$, apply the quotient of powers rule and simplify the expression.
Q: What are some common mistakes to avoid when working with exponents? A: Some common mistakes to avoid when working with exponents include:
- Forgetting to flip the fraction when rewriting a negative exponent as a positive exponent
- Not applying the correct rule of exponents
- Not simplifying the expression after applying the rules of exponents
Q: How can I practice working with exponents and equivalent expressions? A: You can practice working with exponents and equivalent expressions by:
- Solving problems and exercises in a textbook or online resource
- Creating your own problems and exercises
- Working with a tutor or teacher to review and practice the concepts
Q: What are some real-world applications of exponents and equivalent expressions? A: Exponents and equivalent expressions have many real-world applications, including:
- Science: Exponents are used to represent repeated multiplication of a number, which is important in scientific calculations.
- Engineering: Exponents are used to represent repeated multiplication of a number, which is important in engineering calculations.
- Finance: Exponents are used to represent repeated multiplication of a number, which is important in financial calculations.
Conclusion
In conclusion, exponents and equivalent expressions are important concepts in mathematics that have many real-world applications. By understanding the rules of exponents and how to apply them, you can simplify expressions and solve problems more easily.
Related Topics
- Exponents and powers
- Rules of exponents
- Simplifying expressions
- Equivalent expressions
Practice Problems
- Find the equivalent expression for $2^{-3}$.
- Simplify the expression $\frac{1}{4^2}$.
- Find the equivalent expression for $5^{-2}$.
References
- [1] "Exponents and Powers" by Math Open Reference
- [2] "Rules of Exponents" by Khan Academy
- [3] "Simplifying Expressions" by Purplemath