Simplify The Expression $\left(\frac{1}{4ab}\right)^{-2}$. Assume $a \neq 0, B \neq 0$.A. $-\frac{1}{16a^2b^2}$ B. $\frac{a^2b^2}{4}$ C. $-16a^2b^2$ D. $16a^2b^2$

Introduction

In this article, we will simplify the given expression $\left(\frac{1}{4ab}\right)^{-2}$, assuming that $a \neq 0$ and $b \neq 0$. We will use the properties of exponents and fractions to simplify the expression.

Understanding the Expression

The given expression is $\left(\frac{1}{4ab}\right)^{-2}$. This expression involves a fraction raised to a negative power. To simplify this expression, we need to understand the properties of exponents and fractions.

Properties of Exponents

When a fraction is raised to a negative power, we can rewrite it as the reciprocal of the fraction raised to the positive power. In other words, $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$.

Simplifying the Expression

Using the property of exponents mentioned above, we can rewrite the given expression as:

$$\left(\frac{1}{4ab}\right)^{-2} = \left(\frac{4ab}{1}\right)^2$$

Now, we can simplify the expression by squaring the numerator and denominator:

$$\left(\frac{4ab}{1}\right)^2 = \frac{(4ab)^2}{1^2}$$

Expanding the numerator, we get:

$$\frac{(4ab)^2}{1^2} = \frac{16a^2b^2}{1}$$

Final Answer

The final answer is $\boxed{\frac{16a^2b^2}{1}}$. However, we need to choose the correct answer from the options provided.

Comparing with Options

Let's compare our final answer with the options provided:

A. $-\frac{1}{16a^2b^2}$ B. $\frac{a^2b^2}{4}$ C. $-16a^2b^2$ D. $16a^2b^2$

Our final answer is $\frac{16a^2b^2}{1}$, which is equivalent to $16a^2b^2$. Therefore, the correct answer is:

The Correct Answer is D. $16a^2b^2$

Conclusion

In this article, we simplified the expression $\left(\frac{1}{4ab}\right)^{-2}$ using the properties of exponents and fractions. We rewrote the expression as the reciprocal of the fraction raised to the positive power and then simplified it by squaring the numerator and denominator. Our final answer is $16a^2b^2$, which is equivalent to option D.

Additional Tips and Tricks

When simplifying expressions involving fractions and exponents, it's essential to remember the following tips and tricks:

• Use the properties of exponents to rewrite the expression in a more manageable form.
• Simplify the expression by squaring the numerator and denominator.
• Compare the final answer with the options provided to choose the correct answer.

Introduction

In our previous article, we simplified the expression $\left(\frac{1}{4ab}\right)^{-2}$ using the properties of exponents and fractions. In this article, we will provide a Q&A guide to help you understand the concept better.

Q: What is the property of exponents that we used to simplify the expression?

A: The property of exponents that we used is $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$. This property allows us to rewrite a fraction raised to a negative power as the reciprocal of the fraction raised to the positive power.

Q: How do we simplify the expression $\left(\frac{1}{4ab}\right)^{-2}$ using the property of exponents?

A: To simplify the expression, we can rewrite it as $\left(\frac{4ab}{1}\right)^2$. Then, we can simplify the expression by squaring the numerator and denominator.

Q: What is the final answer to the expression $\left(\frac{1}{4ab}\right)^{-2}$?

A: The final answer to the expression is $\frac{16a^2b^2}{1}$, which is equivalent to $16a^2b^2$.

Q: How do we choose the correct answer from the options provided?

A: To choose the correct answer, we need to compare our final answer with the options provided. In this case, our final answer is $16a^2b^2$, which is equivalent to option D.

Q: What are some additional tips and tricks for simplifying expressions involving fractions and exponents?

A: Some additional tips and tricks for simplifying expressions involving fractions and exponents include:

• Using the properties of exponents to rewrite the expression in a more manageable form.
• Simplifying the expression by squaring the numerator and denominator.
• Comparing the final answer with the options provided to choose the correct answer.

Q: What are some common mistakes to avoid when simplifying expressions involving fractions and exponents?

A: Some common mistakes to avoid when simplifying expressions involving fractions and exponents include:

• Not using the properties of exponents to rewrite the expression in a more manageable form.
• Not simplifying the expression by squaring the numerator and denominator.
• Not comparing the final answer with the options provided to choose the correct answer.

Q: How can I practice simplifying expressions involving fractions and exponents?

A: You can practice simplifying expressions involving fractions and exponents by working through examples and exercises. You can also try simplifying expressions on your own and then checking your answers with the correct solutions.

Conclusion

In this article, we provided a Q&A guide to help you understand the concept of simplifying expressions involving fractions and exponents. We covered topics such as the property of exponents, simplifying the expression, and choosing the correct answer. We also provided additional tips and tricks for simplifying expressions involving fractions and exponents. By following these tips and tricks, you can simplify expressions involving fractions and exponents with ease.

Additional Resources

If you want to learn more about simplifying expressions involving fractions and exponents, you can check out the following resources:

• Khan Academy: Simplifying Expressions Involving Fractions and Exponents
• Mathway: Simplifying Expressions Involving Fractions and Exponents
• Wolfram Alpha: Simplifying Expressions Involving Fractions and Exponents

By using these resources, you can get more practice and improve your skills in simplifying expressions involving fractions and exponents.