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

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Understanding the Problem

The given problem involves simplifying an expression with a negative exponent. We are required to simplify the expression $\left(\frac{1}{4ab}\right)^{-2}$, assuming that $a \neq 0$ and $b \neq 0$. This means we need to apply the rules of exponents to simplify the expression.

Applying the Rules of Exponents

To simplify the expression, we need to apply the rule for negative exponents, which states that $a^{-n} = \frac{1}{a^n}$. In this case, we have $\left(\frac{1}{4ab}\right)^{-2}$, which can be rewritten as $\frac{1}{\left(\frac{1}{4ab}\right)^2}$.

Simplifying the Expression

Now, we need to simplify the expression inside the parentheses. We can do this by applying the rule for squaring a fraction, which states that $\left(\frac{a}{b}\right)^2 = \frac{a^2}{b^2}$. In this case, we have $\left(\frac{1}{4ab}\right)^2$, which can be rewritten as $\frac{1^2}{(4ab)^2}$.

Further Simplification

Now, we can simplify the expression further by applying the rule for squaring a product, which states that $(ab)^2 = a^2b^2$. In this case, we have $(4ab)^2$, which can be rewritten as $4^2a^2b^2$.

Final Simplification

Now, we can simplify the expression further by applying the rule for squaring a number, which states that $a^2 = a \cdot a$. In this case, we have $4^2$, which can be rewritten as $4 \cdot 4$.

Combining the Results

Now, we can combine the results of the previous steps to simplify the expression. We have $\frac{1^2}{(4ab)^2} = \frac{1}{4^2a^2b^2} = \frac{1}{16a^2b^2}$.

Conclusion

In conclusion, the simplified expression is $\frac{1}{16a^2b^2}$. This is the correct answer.

Comparison with the Options

Now, let's compare the simplified expression with the options provided.

• Option A: $-16a^2b^2$ is incorrect because it has a negative sign, but the simplified expression does not have a negative sign.
• Option B: $\frac{a^2b^2}{4}$ is incorrect because it has a positive sign in the numerator, but the simplified expression has a negative sign in the denominator.
• Option C: $-\frac{1}{16a^2b^2}$ is correct because it has the same negative sign and the same denominator as the simplified expression.
• Option D: $16a^2b^2$ is incorrect because it has a positive sign, but the simplified expression has a negative sign.

Final Answer

The final answer is $\boxed{C}$.

Discussion

This problem involves simplifying an expression with a negative exponent. The key concept is to apply the rules of exponents to simplify the expression. The rule for negative exponents states that $a^{-n} = \frac{1}{a^n}$. In this case, we have $\left(\frac{1}{4ab}\right)^{-2}$, which can be rewritten as $\frac{1}{\left(\frac{1}{4ab}\right)^2}$. We can then simplify the expression inside the parentheses by applying the rule for squaring a fraction. Finally, we can simplify the expression further by applying the rule for squaring a product and the rule for squaring a number.

Tips and Tricks

• When simplifying expressions with negative exponents, it's essential to apply the rule for negative exponents.
• When simplifying expressions with fractions, it's essential to apply the rule for squaring a fraction.
• When simplifying expressions with products, it's essential to apply the rule for squaring a product.
• When simplifying expressions with numbers, it's essential to apply the rule for squaring a number.

Practice Problems

• Simplify the expression $\left(\frac{1}{2x}\right)^{-3}$.
• Simplify the expression $\left(\frac{1}{3y}\right)^{-2}$.
• Simplify the expression $\left(\frac{1}{4z}\right)^{-4}$.

Conclusion

In conclusion, simplifying expressions with negative exponents requires applying the rules of exponents. The key concept is to apply the rule for negative exponents, which states that $a^{-n} = \frac{1}{a^n}$. By applying this rule and other rules of exponents, we can simplify expressions and arrive at the correct answer.

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Q: What is the rule for negative exponents?

A: The rule for negative exponents states 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.

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

A: To simplify the expression, we need to apply the rule for negative exponents. We can rewrite the expression as $\frac{1}{\left(\frac{1}{4ab}\right)^2}$.

Q: How do we simplify the expression inside the parentheses?

A: To simplify the expression inside the parentheses, we need to apply the rule for squaring a fraction. We can rewrite the expression as $\frac{1^2}{(4ab)^2}$.

Q: How do we simplify the expression further?

A: To simplify the expression further, we need to apply the rule for squaring a product. We can rewrite the expression as $\frac{1}{4^2a^2b^2}$.

Q: How do we simplify the expression even further?

A: To simplify the expression even further, we need to apply the rule for squaring a number. We can rewrite the expression as $\frac{1}{16a^2b^2}$.

Q: What is the final simplified expression?

A: The final simplified expression is $\frac{1}{16a^2b^2}$.

Q: How do we compare the simplified expression with the options provided?

A: To compare the simplified expression with the options provided, we need to look at the signs and the denominators. The simplified expression has a negative sign and a denominator of $16a^2b^2$. We can then compare this with the options provided.

Q: Which option is correct?

A: The correct option is $\boxed{C}$, which is $-\frac{1}{16a^2b^2}$.

Q: What are some tips and tricks for simplifying expressions with negative exponents?

A: Some tips and tricks for simplifying expressions with negative exponents include:

• Applying the rule for negative exponents
• Applying the rule for squaring a fraction
• Applying the rule for squaring a product
• Applying the rule for squaring a number

Q: What are some practice problems for simplifying expressions with negative exponents?

A: Some practice problems for simplifying expressions with negative exponents include:

• Simplify the expression $\left(\frac{1}{2x}\right)^{-3}$
• Simplify the expression $\left(\frac{1}{3y}\right)^{-2}$
• Simplify the expression $\left(\frac{1}{4z}\right)^{-4}$

Q: What is the importance of simplifying expressions with negative exponents?

A: The importance of simplifying expressions with negative exponents is that it allows us to rewrite the expression in a more convenient form. This can make it easier to work with the expression and to arrive at the correct answer.

Q: How do we know when to apply the rule for negative exponents?

A: We know when to apply the rule for negative exponents when we see a negative exponent in the expression. This is a clear indication that we need to apply the rule for negative exponents.

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

A: Some common mistakes to avoid when simplifying expressions with negative exponents include:

• Not applying the rule for negative exponents
• Applying the rule for negative exponents incorrectly
• Not simplifying the expression further

Q: How do we check our work when simplifying expressions with negative exponents?

A: We can check our work by plugging the simplified expression back into the original expression and verifying that it is true. This can help us to ensure that we have simplified the expression correctly.