The given expression is:

$\left(-\frac{3}{8}\right)^4 \times \left(-\frac{3}{8}\right)^{-9}$

Step 1: Apply the rule of exponents

When multiplying terms with the same base, you add their exponents:

$\left(-\frac{3}{8}\right)^4 \times \left(-\frac{3}{8}\right)^{-9} = \left(-\frac{3}{8}\right)^{4 + (-9)}$

Step 2: Simplify the exponent

$4 + (-9) = -5$

So, the expression becomes: $\left(-\frac{3}{8}\right)^{-5}$

Step 3: Simplify the negative exponent

A negative exponent indicates the reciprocal: $\left(-\frac{3}{8}\right)^{-5} = \frac{1}{\left(-\frac{3}{8}\right)^5}$

Step 4: Calculate the power

To simplify $\left(-\frac{3}{8}\right)^5$, raise both the numerator and denominator to the power of 5: $\left(-\frac{3}{8}\right)^5 = \frac{(-3)^5}{8^5} = \frac{-243}{32768}$

Thus: $\frac{1}{\left(-\frac{3}{8}\right)^5} = \frac{1}{\frac{-243}{32768}} = \frac{32768}{-243}$

Simplify the sign: $\frac{32768}{-243} = -\frac{32768}{243}$

Final Answer:

$-\frac{32768}{243}$

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Would you like more details about any step?

Expanding your understanding:

1. How do negative exponents work in different contexts?
2. What are some real-world applications of exponent rules?
3. How can you verify calculations involving large powers?
4. Why does the sign of the base matter in exponentiation?
5. How do fractional bases behave under odd and even exponents?

Tip: When dealing with powers and fractions, simplify early to avoid large numbers!