What Is The Value Of ( − 3 4 ) − 4 \left(-\frac{3}{4}\right)^{-4} ( − 4 3 ​ ) − 4 ?A. − 256 81 -\frac{256}{81} − 81 256 ​ B. − 81 256 -\frac{81}{256} − 256 81 ​ C. 81 256 \frac{81}{256} 256 81 ​ D. 258 81 \frac{258}{81} 81 258 ​

When dealing with negative exponents, it's essential to understand the concept of reciprocals and how they relate to the original expression. In this article, we'll delve into the world of negative exponents and explore the value of $\left(-\frac{3}{4}\right)^{-4}$.

What are Negative Exponents?

Negative exponents are a way of expressing a fraction in a different form. When we have a negative exponent, it's equivalent to taking the reciprocal of the base and changing the sign of the exponent. In other words, $a^{-n} = \frac{1}{a^n}$.

Applying the Concept to the Given Expression

Now that we understand the concept of negative exponents, let's apply it to the given expression: $\left(-\frac{3}{4}\right)^{-4}$. Using the definition of negative exponents, we can rewrite this expression as:

$$\left(-\frac{3}{4}\right)^{-4} = \frac{1}{\left(-\frac{3}{4}\right)^4}$$

Simplifying the Expression

To simplify the expression, we need to evaluate the exponent of the base. When we raise a fraction to a power, we raise both the numerator and the denominator to that power. In this case, we have:

$$\left(-\frac{3}{4}\right)^4 = \left(\frac{-3}{4}\right)^4 = \frac{(-3)^4}{4^4}$$

Evaluating the Numerator and Denominator

Now that we have the expression in a simpler form, let's evaluate the numerator and denominator separately:

$$(-3)^4 = 81$$

$$4^4 = 256$$

Putting it All Together

Now that we have the values of the numerator and denominator, we can put them together to get the final result:

$$\frac{1}{\left(-\frac{3}{4}\right)^4} = \frac{1}{\frac{81}{256}} = \frac{256}{81}$$

Conclusion

In conclusion, the value of $\left(-\frac{3}{4}\right)^{-4}$ is $\frac{256}{81}$. This result is obtained by applying the concept of negative exponents and simplifying the expression using the rules of exponents.

Comparison with the Given Options

Now that we have the final result, let's compare it with the given options:

• A. $-\frac{256}{81}$: This is not the correct result.
• B. $-\frac{81}{256}$: This is not the correct result.
• C. $\frac{81}{256}$: This is not the correct result.
• D. $\frac{256}{81}$: This is the correct result.

Final Answer

$$

In this section, we'll address some common questions related to negative exponents and the value of $\left(-\frac{3}{4}\right)^{-4}$.

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

A: A positive exponent indicates that the base is raised to that power, while a negative exponent indicates that the reciprocal of the base is raised to that power.

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

A: To simplify an expression with a negative exponent, you can rewrite it as the reciprocal of the base raised to the positive power of the exponent.

Q: Can you provide an example of simplifying an expression with a negative exponent?

A: Let's consider the expression $\left(-\frac{3}{4}\right)^{-4}$. To simplify this expression, we can rewrite it as:

$$\left(-\frac{3}{4}\right)^{-4} = \frac{1}{\left(-\frac{3}{4}\right)^4}$$

Q: How do I evaluate the exponent of a fraction?

A: When evaluating the exponent of a fraction, you need to raise both the numerator and the denominator to that power.

Q: Can you provide an example of evaluating the exponent of a fraction?

A: Let's consider the expression $\left(-\frac{3}{4}\right)^4$. To evaluate this expression, we can raise both the numerator and the denominator to the power of 4:

$$\left(-\frac{3}{4}\right)^4 = \left(\frac{-3}{4}\right)^4 = \frac{(-3)^4}{4^4}$$

Q: How do I simplify the expression $\frac{(-3)^4}{4^4}$?

A: To simplify this expression, we can evaluate the numerator and denominator separately:

$$(-3)^4 = 81$$

$$4^4 = 256$$

Q: What is the final result of the expression $\left(-\frac{3}{4}\right)^{-4}$?

A: The final result of the expression $\left(-\frac{3}{4}\right)^{-4}$ is $\frac{256}{81}$.

Q: Can you provide a summary of the steps to simplify an expression with a negative exponent?

A: Here's a summary of the steps to simplify an expression with a negative exponent:

1. Rewrite the expression as the reciprocal of the base raised to the positive power of the exponent.
2. Evaluate the exponent of the base by raising both the numerator and the denominator to that power.
3. Simplify the expression by evaluating the numerator and denominator separately.
4. Combine the results to get the final answer.

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

A: Here are some common mistakes to avoid when simplifying expressions with negative exponents:

• Not rewriting the expression as the reciprocal of the base raised to the positive power of the exponent.
• Not evaluating the exponent of the base correctly.
• Not simplifying the expression by evaluating the numerator and denominator separately.
• Not combining the results correctly to get the final answer.

Q: Can you provide additional examples of simplifying expressions with negative exponents?

A: Here are some additional examples of simplifying expressions with negative exponents:

• $\left(-\frac{2}{3}\right)^{-3} = \frac{1}{\left(-\frac{2}{3}\right)^3} = \frac{1}{\frac{(-2)^3}{3^3}} = \frac{3^3}{(-2)^3} = \frac{27}{-8} = -\frac{27}{8}$
• $\left(\frac{1}{2}\right)^{-2} = \frac{1}{\left(\frac{1}{2}\right)^2} = \frac{1}{\frac{1^2}{2^2}} = \frac{2^2}{1^2} = \frac{4}{1} = 4$

Q: Can you provide a list of resources for learning more about negative exponents and simplifying expressions?

A: Here are some resources for learning more about negative exponents and simplifying expressions:

• Khan Academy: Negative Exponents
• Mathway: Simplifying Expressions with Negative Exponents
• Wolfram Alpha: Negative Exponents
• MIT OpenCourseWare: Algebra
• Khan Academy: Algebra

Conclusion

In conclusion, simplifying expressions with negative exponents requires a clear understanding of the concept of negative exponents and the rules for simplifying expressions. By following the steps outlined in this article, you can simplify expressions with negative exponents and arrive at the correct solution.