Calculating Negative Exponents

Oct 13, 2025 by magicalceylontours.com 98 views

When dealing with exponents, it's essential to understand the concept of negative exponents. A negative exponent is a shorthand way of expressing a fraction with a positive exponent. In this article, we will explore the value of $\left(-\frac{3}{4}\right)^{-4}$ and how to calculate it.

What is a Negative Exponent?

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 number raised to a positive power. For example, $a^{-n} = \frac{1}{a^n}$ .

Calculating Negative Exponents

To calculate a negative exponent, we can use the following formula:

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

This formula states that a number raised to a negative power is equal to the reciprocal of the number raised to a positive power.

Applying the Formula to the Given Problem

Now that we understand the concept of negative exponents, let's apply the formula to the given problem:

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

Simplifying the Expression

To simplify the expression, we need to calculate the value of $\left(-\frac{3}{4}\right)^4$ . We can do this by raising the fraction to the power of 4:

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

Multiplying the Fractions

To multiply the fractions, we need to multiply the numerators and denominators separately:

$\left(-\frac{3}{4}\right) \times \left(-\frac{3}{4}\right) = \frac{9}{16}$

$\frac{9}{16} \times \left(-\frac{3}{4}\right) = \frac{-27}{64}$

$\frac{-27}{64} \times \left(-\frac{3}{4}\right) = \frac{81}{256}$

Reciprocal of the Result

Now that we have calculated the value of $\left(-\frac{3}{4}\right)^4$ , we can find the reciprocal of the result:

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

Simplifying the Reciprocal

To simplify the reciprocal, we can multiply the numerator and denominator by the reciprocal of the denominator:

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

Conclusion

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

Answer

The correct answer is:

D. $\frac{256}{81}$

Final Thoughts

$Understanding the concept of negative exponents is crucial in mathematics, as it allows us to simplify complex expressions and solve problems more efficiently. By applying the formula for negative exponents and simplifying the expression, we can find the value of $\left(-\frac{3}{4}\right)^{-4}$ and other similar expressions.<br/> **Frequently Asked Questions (FAQs)**$

In this section, we will address some of the most common questions related to negative exponents and their applications.

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

A: A positive exponent is a number raised to a power that is greater than zero, while a negative exponent is a number raised to a power that is less than zero. For example, $a^2$ is a positive exponent, while $a^{-2}$ is a negative exponent.

Q: How do I calculate a negative exponent?

A: To calculate a negative exponent, you can use the formula $a^{-n} = \frac{1}{a^n}$ . This formula states that a number raised to a negative power is equal to the reciprocal of the number raised to a positive power.

Q: What is the reciprocal of a number?

A: The reciprocal of a number is 1 divided by that number. For example, the reciprocal of 2 is $\frac{1}{2}$ , and the reciprocal of $-\frac{3}{4}$ is $-\frac{4}{3}$ .

Q: Can I simplify a negative exponent expression?

A: Yes, you can simplify a negative exponent expression by applying the formula $a^{-n} = \frac{1}{a^n}$ . This will allow you to rewrite the expression in a simpler form.

Q: How do I apply the formula for negative exponents to a problem?

A: To apply the formula for negative exponents to a problem, you need to follow these steps:

Identify the negative exponent in the problem.
Rewrite the expression using the formula $a^{-n} = \frac{1}{a^n}$ .
Simplify the resulting expression.

Q: What are some common applications of negative exponents?

A: Negative exponents have many applications in mathematics, including:

Simplifying complex expressions
Solving equations
Calculating probabilities
Working with fractions and decimals

Q: Can I use a calculator to calculate a negative exponent?

A: Yes, you can use a calculator to calculate a negative exponent. However, it's essential to understand the concept of negative exponents and how to apply the formula to simplify the expression.

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 apply the formula $a^{-n} = \frac{1}{a^n}$ .
Not simplifying the resulting expression.
Using the wrong formula or method.

Conclusion

In conclusion, negative exponents are a fundamental concept in mathematics that can be used to simplify complex expressions and solve problems more efficiently. By understanding the concept of negative exponents and how to apply the formula, you can become more confident and proficient in your math skills.

Additional Resources

If you're looking for more information on negative exponents or want to practice your skills, here are some additional resources you can use:

Online math tutorials and videos
Math textbooks and workbooks
Online calculators and tools
Math apps and software

Final Thoughts

Remember, practice makes perfect. The more you practice working with negative exponents, the more comfortable you'll become with the concept and the more confident you'll be in your math skills.