To simplify the expression $(4 \cdot 10^{-5})^3 / (4 \cdot 10^4)^3$ using the rules of exponents, we will follow these steps:

1. Apply the Power Rule: $(a \cdot b)^n = a^n \cdot b^n$

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

3. Express the final answer with positive exponents.

Step 1: Apply the Power Rule

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

$(4 \cdot 10^4)^3 = 4^3 \cdot (10^4)^3$

Step 2: Compute the powers

$4^3 = 64$

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

$(10^4)^3 = 10^{12}$

Now the expression becomes:

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

Step 3: Simplify the fraction

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

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

To express the answer with positive exponents:

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

Final Answer:

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

Would you like any further clarifications or have any questions?

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Relative Questions

1. What are the rules for multiplying and dividing powers with the same base?
2. How do you convert a negative exponent to a positive exponent?
3. How does raising a product to a power work in terms of individual factors?
4. Can you show an example where the base is negative and the exponent is an odd number?
5. How do you handle calculations when the base is a fraction?
6. What are the differences between positive and negative exponents?
7. How would the expression change if the exponent outside the parentheses was different?
8. What are the potential pitfalls when simplifying complex exponential expressions?

Tip:

When simplifying expressions with exponents, make sure to consistently apply the rules and carefully handle both positive and negative exponents to avoid errors.