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. 256 81 \frac{256}{81} 81 256 ​

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 explore the value of $\left(-\frac{3}{4}\right)^{-4}$ and provide a step-by-step solution to determine the correct answer.

What are Negative Exponents?

A negative exponent is a mathematical operation that involves raising a number to a power that is less than zero. In other words, it's the reciprocal of the positive exponent. For example, $a^{-n} = \frac{1}{a^n}$. This means that when we have a negative exponent, we can rewrite it as a fraction with the reciprocal of the base and the positive exponent.

Properties of Negative Exponents

To solve the problem $\left(-\frac{3}{4}\right)^{-4}$, we need to understand the properties of negative exponents. One of the key properties is that when we have a negative exponent, we can rewrite it as a fraction with the reciprocal of the base and the positive exponent. This property can be expressed as:

$a^{-n} = \frac{1}{a^n}$

Step-by-Step Solution

Now that we have a good understanding of negative exponents, let's apply this knowledge to solve the problem $\left(-\frac{3}{4}\right)^{-4}$.

Step 1: Rewrite the Negative Exponent

Using the property of negative exponents, we can rewrite $\left(-\frac{3}{4}\right)^{-4}$ as:

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

Step 2: Simplify the Expression

Now that we have rewritten the negative exponent, let's simplify the expression. We can start by evaluating the exponent:

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

Step 3: Take the Reciprocal

Now that we have evaluated the exponent, let's take the reciprocal of the result:

$\frac{1}{\frac{81}{256}} = \frac{256}{81}$

Conclusion

In conclusion, the value of $\left(-\frac{3}{4}\right)^{-4}$ is $\frac{256}{81}$. This is the correct answer, and it's essential to understand the properties of negative exponents to arrive at this solution.

Comparison of Options

Let's compare our solution with the given options:

• A. $-\frac{256}{81}$: This is incorrect, as the negative sign is not present in our solution.
• B. $-\frac{81}{256}$: This is incorrect, as the negative sign is not present in our solution.
• C. $\frac{81}{256}$: This is incorrect, as the order of the numerator and denominator is reversed in our solution.
• D. $\frac{256}{81}$: This is the correct answer.

Final Thoughts

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Q: What is the rule for negative exponents?

A: The rule for negative exponents is that $a^{-n} = \frac{1}{a^n}$. This means that when we have a negative exponent, we can rewrite it as a fraction with the reciprocal of the base and the positive exponent.

Q: How do I simplify expressions with negative exponents?

A: To simplify expressions with negative exponents, we can follow these steps:

1. Rewrite the negative exponent as a fraction with the reciprocal of the base and the positive exponent.
2. Simplify the expression by evaluating the exponent.
3. Take the reciprocal of the result.

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

A: A negative exponent is a mathematical operation that involves raising a number to a power that is less than zero. A positive exponent, on the other hand, involves raising a number to a power that is greater than zero. For example, $a^{-n}$ is the reciprocal of $a^n$.

Q: Can I use a calculator to evaluate expressions with negative exponents?

A: Yes, you can use a calculator to evaluate expressions with negative exponents. However, it's essential to understand the properties of negative exponents and how to simplify expressions before using a calculator.

Q: How do I determine the value of an expression with a negative exponent?

A: To determine the value of an expression with a negative exponent, you can follow these steps:

1. Rewrite the negative exponent as a fraction with the reciprocal of the base and the positive exponent.
2. Simplify the expression by evaluating the exponent.
3. Take the reciprocal of the result.

Q: Can I use the order of operations to evaluate expressions with negative exponents?

A: Yes, you can use the order of operations to evaluate expressions with negative exponents. However, it's essential to remember that negative exponents have a higher precedence than positive exponents.

Q: What is the relationship between negative exponents and fractions?

A: Negative exponents and fractions are closely related. When we have a negative exponent, we can rewrite it as a fraction with the reciprocal of the base and the positive exponent.

Q: Can I use negative exponents to simplify complex expressions?

A: Yes, you can use negative exponents to simplify complex expressions. By rewriting negative exponents as fractions, you can simplify expressions and make them easier to evaluate.

Q: How do I apply the properties of negative exponents to solve problems?

A: To apply the properties of negative exponents to solve problems, you can follow these steps:

1. Identify the negative exponent in the expression.
2. Rewrite the negative exponent as a fraction with the reciprocal of the base and the positive exponent.
3. Simplify the expression by evaluating the exponent.
4. Take the reciprocal of the result.

Q: Can I use negative exponents to solve equations?

A: Yes, you can use negative exponents to solve equations. By applying the properties of negative exponents, you can simplify equations and make them easier to solve.

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:

• Forgetting to rewrite negative exponents as fractions
• Not simplifying expressions correctly
• Not taking the reciprocal of the result
• Not applying the order of operations correctly

Q: How can I practice working with negative exponents?

A: You can practice working with negative exponents by:

• Solving problems that involve negative exponents
• Simplifying expressions with negative exponents
• Applying the properties of negative exponents to solve equations
• Using online resources and practice exercises to reinforce your understanding of negative exponents.