Simplify The Expression: $\left(4 B^{-4} C^6\right)^{-3}$ Write Your Answer Using Only Positive Exponents.

Understanding the Problem

When dealing with exponents, it's essential to understand the rules that govern their behavior. In this problem, we're given the expression $\left(4 b^{-4} c^6\right)^{-3}$ and asked to simplify it using only positive exponents. To approach this problem, we need to apply the rules of exponents, specifically the power rule and the product rule.

Applying the Power Rule

The power rule states that for any variables $a$ and $b$ and any integers $m$ and $n$, we have:

$$\left(a^m\right)^n = a^{m \cdot n}$$

Using this rule, we can simplify the expression $\left(4 b^{-4} c^6\right)^{-3}$ by applying the exponent $-3$ to each of the factors inside the parentheses.

Simplifying the Expression

Applying the power rule, we get:

$$\left(4 b^{-4} c^6\right)^{-3} = 4^{-3} \left(b^{-4}\right)^{-3} \left(c^6\right)^{-3}$$

Now, we can simplify each of the factors using the power rule:

$$4^{-3} = \frac{1}{4^3} = \frac{1}{64}$$

$$\left(b^{-4}\right)^{-3} = b^{4 \cdot 3} = b^{12}$$

$$\left(c^6\right)^{-3} = c^{-6 \cdot 3} = c^{-18}$$

Combining the Factors

Now that we've simplified each of the factors, we can combine them to get the final expression:

$$\frac{1}{64} b^{12} c^{-18}$$

Writing the Answer with Positive Exponents

To write the answer using only positive exponents, we need to get rid of the negative exponent $c^{-18}$. We can do this by taking the reciprocal of the factor and changing the sign of the exponent:

$$\frac{1}{64} b^{12} c^{-18} = \frac{1}{64} b^{12} \frac{1}{c^{18}}$$

Final Answer

The final answer is $\boxed{\frac{b^{12}}{64c^{18}}}$.

Understanding the Problem

When dealing with exponents, it's essential to understand the rules that govern their behavior. In this problem, we're given the expression $\left(4 b^{-4} c^6\right)^{-3}$ and asked to simplify it using only positive exponents. To approach this problem, we need to apply the rules of exponents, specifically the power rule and the product rule.

Q&A

Q: What is the power rule in exponents?

A: The power rule states that for any variables $a$ and $b$ and any integers $m$ and $n$, we have:

$$\left(a^m\right)^n = a^{m \cdot n}$$

This rule allows us to simplify expressions by applying an exponent to each of the factors inside the parentheses.

Q: How do I apply the power rule to the given expression?

A: To apply the power rule, we need to multiply the exponents of each of the factors inside the parentheses. In this case, we have:

$$\left(4 b^{-4} c^6\right)^{-3} = 4^{-3} \left(b^{-4}\right)^{-3} \left(c^6\right)^{-3}$$

Q: What is the product rule in exponents?

A: The product rule states that for any variables $a$ and $b$ and any integers $m$ and $n$, we have:

$$a^m \cdot a^n = a^{m + n}$$

This rule allows us to simplify expressions by combining the exponents of the same base.

Q: How do I simplify the expression using the product rule?

A: To simplify the expression using the product rule, we need to combine the exponents of the same base. In this case, we have:

$$4^{-3} \left(b^{-4}\right)^{-3} \left(c^6\right)^{-3} = \frac{1}{4^3} b^{12} c^{-18}$$

Q: How do I write the answer using only positive exponents?

A: To write the answer using only positive exponents, we need to get rid of the negative exponent $c^{-18}$. We can do this by taking the reciprocal of the factor and changing the sign of the exponent:

$$\frac{1}{4^3} b^{12} c^{-18} = \frac{1}{4^3} b^{12} \frac{1}{c^{18}}$$

Q: What is the final answer?

A: The final answer is $\boxed{\frac{b^{12}}{64c^{18}}}$.

Common Mistakes to Avoid

When simplifying expressions using exponents, it's essential to avoid common mistakes. Here are a few to watch out for:

• Not applying the power rule correctly: Make sure to multiply the exponents of each of the factors inside the parentheses.
• Not combining the exponents correctly: Make sure to combine the exponents of the same base using the product rule.
• Not getting rid of negative exponents: Make sure to take the reciprocal of the factor and change the sign of the exponent to get rid of negative exponents.

Conclusion

Simplifying expressions using exponents can be a challenging task, but with practice and patience, you can master the rules and techniques. Remember to apply the power rule and the product rule correctly, and don't forget to get rid of negative exponents. With these tips and tricks, you'll be well on your way to simplifying expressions like a pro!