What Is The Value Of $\left(-\frac{3}{4}\right)^{-4}$?A. $-\frac{256}{81}$B. $-\frac{81}{256}$C. $\frac{81}{256}$D. $\frac{256}{81}$

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Understanding the Concept of Negative Exponents

When dealing with negative exponents, we need to understand that they represent the reciprocal of the base raised to the positive exponent. In other words, (ab)−n=(ba)n\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n. This concept is crucial in solving the given problem.

Applying the Concept to the Given Problem

To find the value of (−34)−4\left(-\frac{3}{4}\right)^{-4}, we can use the concept of negative exponents. We can rewrite the expression as (4−3)4\left(\frac{4}{-3}\right)^4. This simplifies the problem and allows us to apply the exponent rule.

Applying the Exponent Rule

When raising a fraction to a power, we raise the numerator and the denominator to that power. In this case, we have (4−3)4=44(−3)4\left(\frac{4}{-3}\right)^4 = \frac{4^4}{(-3)^4}. Now, we can simplify the expression by evaluating the numerator and the denominator.

Evaluating the Numerator and the Denominator

The numerator is 444^4, which equals 256256. The denominator is (−3)4(-3)^4, which equals 8181. Therefore, the expression simplifies to 25681\frac{256}{81}.

Conclusion

Based on the concept of negative exponents and the exponent rule, we can conclude that the value of (−34)−4\left(-\frac{3}{4}\right)^{-4} is 25681\frac{256}{81}.

Final Answer

The final answer is 25681\boxed{\frac{256}{81}}.

Discussion

The given problem requires a deep understanding of the concept of negative exponents and the exponent rule. By applying these concepts, we can simplify the expression and arrive at the correct answer. The problem is a great example of how mathematical concepts can be applied to solve real-world problems.

Related Problems

  • (23)−2\left(\frac{2}{3}\right)^{-2}
  • (−12)−3\left(-\frac{1}{2}\right)^{-3}
  • (34)−1\left(\frac{3}{4}\right)^{-1}

Tips and Tricks

  • When dealing with negative exponents, remember that they represent the reciprocal of the base raised to the positive exponent.
  • When raising a fraction to a power, raise the numerator and the denominator to that power.
  • Simplify the expression by evaluating the numerator and the denominator.

Common Mistakes

  • Failing to recognize the concept of negative exponents
  • Not applying the exponent rule correctly
  • Not simplifying the expression properly

Real-World Applications

  • The concept of negative exponents has numerous real-world applications in fields such as physics, engineering, and computer science.
  • Understanding the exponent rule is crucial in solving problems related to finance, economics, and data analysis.

Conclusion

In conclusion, the value of (−34)−4\left(-\frac{3}{4}\right)^{-4} is 25681\frac{256}{81}. By applying the concept of negative exponents and the exponent rule, we can simplify the expression and arrive at the correct answer. The problem is a great example of how mathematical concepts can be applied to solve real-world problems.

Frequently Asked Questions

Q: What is a negative exponent?

A: A negative exponent represents the reciprocal of the base raised to the positive exponent. In other words, (ab)−n=(ba)n\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n.

Q: How do I apply the exponent rule when dealing with negative exponents?

A: To apply the exponent rule, simply rewrite the negative exponent as a positive exponent by taking the reciprocal of the base. For example, (−34)−4=(4−3)4\left(-\frac{3}{4}\right)^{-4} = \left(\frac{4}{-3}\right)^4.

Q: What is the difference between a negative exponent and a positive exponent?

A: A negative exponent represents the reciprocal of the base raised to the positive exponent, while a positive exponent represents the base raised to that power. For example, (34)−2=(43)2\left(\frac{3}{4}\right)^{-2} = \left(\frac{4}{3}\right)^2.

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

A: To simplify an expression with a negative exponent, apply the exponent rule by taking the reciprocal of the base and raising it to the positive exponent. For example, (−34)−4=(4−3)4=25681\left(-\frac{3}{4}\right)^{-4} = \left(\frac{4}{-3}\right)^4 = \frac{256}{81}.

Q: What are some common mistakes to avoid when dealing with negative exponents?

A: Some common mistakes to avoid include:

  • Failing to recognize the concept of negative exponents
  • Not applying the exponent rule correctly
  • Not simplifying the expression properly

Q: How do I apply the exponent rule when dealing with fractions?

A: When dealing with fractions, apply the exponent rule by raising the numerator and the denominator to the power. For example, (23)−2=(32)2=94\left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^2 = \frac{9}{4}.

Q: What are some real-world applications of negative exponents and exponent rules?

A: Negative exponents and exponent rules have numerous real-world applications in fields such as physics, engineering, and computer science. Understanding these concepts is crucial in solving problems related to finance, economics, and data analysis.

Q: How do I practice and improve my understanding of negative exponents and exponent rules?

A: To practice and improve your understanding of negative exponents and exponent rules, try the following:

  • Practice solving problems with negative exponents and exponent rules
  • Review and understand the concept of negative exponents and exponent rules
  • Apply the concepts to real-world problems and scenarios

Additional Resources

  • Khan Academy: Negative Exponents
  • Mathway: Negative Exponents and Exponent Rules
  • Wolfram Alpha: Negative Exponents and Exponent Rules

Conclusion

In conclusion, understanding negative exponents and exponent rules is crucial in solving problems related to mathematics and real-world applications. By applying the concepts and practicing with problems, you can improve your understanding and become proficient in solving complex problems.