To simplify the given expression $(4 \cdot 10^{-5})^3 / (4 \cdot 10^4)^3$ using the product, quotient, and power rules, we'll proceed as follows:

1. Apply the Power Rule: $(a \cdot b)^n = a^n \cdot b^n$ to both the numerator and the denominator.

$(4 \cdot 10^{-5})^3 = 4^3 \cdot (10^{-5})^3$ $(4 \cdot 10^4)^3 = 4^3 \cdot (10^4)^3$

2. Calculate the individual components:

$4^3 = 64$ $(10^{-5})^3 = 10^{-15}$ $(10^4)^3 = 10^{12}$

So, the expression becomes:

$\frac{64 \cdot 10^{-15}}{64 \cdot 10^{12}}$

3. Simplify the fraction:

The $64$ terms cancel out:

$\frac{64 \cdot 10^{-15}}{64 \cdot 10^{12}} = \frac{10^{-15}}{10^{12}}$

4. Apply the Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$

$\frac{10^{-15}}{10^{12}} = 10^{-15 - 12} = 10^{-27}$

Thus, the simplified expression is:

$10^{-27}$

Since the problem requests the answer with only positive exponents, we'll convert the expression as follows:

$10^{-27} = \frac{1}{10^{27}}$

So, the final answer is:

$\frac{1}{10^{27}}$

Would you like more details or have any questions?

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Relative Questions

1. What is the power rule and how is it applied?
2. How do you simplify expressions with negative exponents?
3. Can you provide another example involving both the product and quotient rules?
4. How does the quotient rule differ from the product rule?
5. What happens when you raise a negative exponent to a power?
6. How do you handle expressions with different bases in exponentiation?
7. What are some common mistakes to avoid when using exponent rules?
8. How does the result change if the initial base is a fraction instead of a whole number?

Tip:

Always double-check the base numbers when simplifying expressions with exponents, as small errors can lead to incorrect results.