Evaluate The Expression: ( 2 3 ) − 1 \left(\frac{2}{3}\right)^{-1} ( 3 2 ​ ) − 1

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Introduction

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 the rules and properties that govern their behavior. In this article, we will evaluate the expression $\left(\frac{2}{3}\right)^{-1}$ and explore the underlying mathematical concepts.

Understanding Negative Exponents

A negative exponent is a shorthand way of expressing a fraction. When we see a negative exponent, it can be rewritten as a fraction with the base in the denominator and the absolute value of the exponent as the numerator. In other words, $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$.

Evaluating the Expression

To evaluate the expression $\left(\frac{2}{3}\right)^{-1}$, we can apply the rule mentioned above. We can rewrite the expression as $\left(\frac{3}{2}\right)^1$. This is because the negative exponent is equivalent to flipping the fraction and changing the sign of the exponent.

Simplifying the Expression

Now that we have rewritten the expression, we can simplify it further. Since the exponent is 1, we can simply evaluate the fraction: $\frac{3}{2}$.

Conclusion

In conclusion, evaluating the expression $\left(\frac{2}{3}\right)^{-1}$ involves understanding the concept of negative exponents and applying the corresponding rule. By rewriting the expression as $\left(\frac{3}{2}\right)^1$ and simplifying it, we arrive at the final answer: $\frac{3}{2}$.

Properties of Negative Exponents

Negative exponents have several properties that are essential to understand. Here are a few key properties:

• Property 1: $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$
• Property 2: $\left(\frac{a}{b}\right)^{-n} = \frac{b^n}{a^n}$
• Property 3: $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{-(-n)}$

Examples of Negative Exponents

Here are a few examples of negative exponents:

• $\left(\frac{2}{3}\right)^{-1} = \left(\frac{3}{2}\right)^1 = \frac{3}{2}$
• $\left(\frac{4}{5}\right)^{-2} = \left(\frac{5}{4}\right)^2 = \frac{25}{16}$
• $\left(\frac{1}{2}\right)^{-3} = \left(\frac{2}{1}\right)^3 = 8$

Applications of Negative Exponents

Negative exponents have numerous applications in mathematics and other fields. Here are a few examples:

• Algebra: Negative exponents are used to simplify expressions and solve equations.
• Calculus: Negative exponents are used to represent inverse functions and derivatives.
• Physics: Negative exponents are used to represent decay rates and exponential functions.

Conclusion

In conclusion, negative exponents are a fundamental concept in mathematics that have numerous applications. By understanding the properties and rules of negative exponents, we can evaluate expressions and simplify complex equations. Whether you're a student, teacher, or professional, mastering negative exponents is essential for success in mathematics and other fields.

Final Thoughts

Negative exponents may seem intimidating at first, but with practice and patience, you can master this concept. Remember to apply the rules and properties of negative exponents to simplify expressions and solve equations. With time and effort, you'll become proficient in evaluating expressions with negative exponents and unlock new possibilities in mathematics and beyond.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Physics for Scientists and Engineers" by Paul A. Tipler

Note: The references provided are for educational purposes only and are not intended to be a comprehensive list of resources.

Introduction

In our previous article, we explored the concept of negative exponents and how to evaluate expressions with them. However, we know that practice makes perfect, and there's no better way to reinforce your understanding than by answering questions and solving problems. In this article, we'll provide a Q&A section to help you solidify your knowledge and become more confident in evaluating expressions with negative exponents.

Q&A

Q1: What is the value of $\left(\frac{3}{4}\right)^{-2}$?

A1: To evaluate this expression, we can apply the rule $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$. Therefore, $\left(\frac{3}{4}\right)^{-2} = \left(\frac{4}{3}\right)^2 = \frac{16}{9}$.

Q2: What is the value of $\left(\frac{2}{5}\right)^{-3}$?

A2: Using the same rule as above, we can rewrite the expression as $\left(\frac{5}{2}\right)^3 = \frac{125}{8}$.

Q3: What is the value of $\left(\frac{1}{2}\right)^{-4}$?

A3: Applying the rule, we get $\left(\frac{2}{1}\right)^4 = 16$.

Q4: What is the value of $\left(\frac{3}{4}\right)^{-1}$?

A4: Using the rule, we can rewrite the expression as $\left(\frac{4}{3}\right)^1 = \frac{4}{3}$.

Q5: What is the value of $\left(\frac{2}{3}\right)^{-2}$?

A5: Applying the rule, we get $\left(\frac{3}{2}\right)^2 = \frac{9}{4}$.

Q6: What is the value of $\left(\frac{1}{3}\right)^{-5}$?

A6: Using the rule, we can rewrite the expression as $\left(\frac{3}{1}\right)^5 = 243$.

Q7: What is the value of $\left(\frac{4}{5}\right)^{-1}$?

A7: Applying the rule, we get $\left(\frac{5}{4}\right)^1 = \frac{5}{4}$.

Q8: What is the value of $\left(\frac{2}{3}\right)^{-3}$?

A8: Using the rule, we can rewrite the expression as $\left(\frac{3}{2}\right)^3 = \frac{27}{8}$.

Q9: What is the value of $\left(\frac{1}{4}\right)^{-2}$?

A9: Applying the rule, we get $\left(\frac{4}{1}\right)^2 = 16$.

Q10: What is the value of $\left(\frac{3}{2}\right)^{-1}$?

A10: Using the rule, we can rewrite the expression as $\left(\frac{2}{3}\right)^1 = \frac{2}{3}$.

Conclusion

In this Q&A section, we've provided 10 questions and answers to help you practice evaluating expressions with negative exponents. By working through these examples, you'll become more confident in applying the rules and properties of negative exponents. Remember to practice regularly to reinforce your understanding and become proficient in evaluating expressions with negative exponents.

Final Thoughts

Evaluating expressions with negative exponents may seem challenging at first, but with practice and patience, you can master this concept. Remember to apply the rules and properties of negative exponents to simplify expressions and solve equations. With time and effort, you'll become proficient in evaluating expressions with negative exponents and unlock new possibilities in mathematics and beyond.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Physics for Scientists and Engineers" by Paul A. Tipler

Note: The references provided are for educational purposes only and are not intended to be a comprehensive list of resources.