Let's simplify the given expression step-by-step. The expression is:

$\left( \left( 8x^{-3} y^9 \right)^{-3} \right)^7$

Step 1: Apply the exponent inside the parentheses

We start by simplifying the inner part $\left( 8x^{-3} y^9 \right)^{-3}$.

Using the rule $(a^m b^n)^p = a^{mp} b^{np}$, we distribute the exponent $-3$ to each factor inside the parentheses:

$8^{-3} \cdot (x^{-3})^{-3} \cdot (y^9)^{-3}$

This simplifies to:

$8^{-3} \cdot x^9 \cdot y^{-27}$

Step 2: Apply the outer exponent

Now, raise the entire expression $\left(8^{-3} x^9 y^{-27}\right)$ to the power of 7:

$\left( 8^{-3} \cdot x^9 \cdot y^{-27} \right)^7$

Using the same rule as before $(a^m b^n)^p = a^{mp} b^{np}$, distribute the exponent 7:

$8^{-21} \cdot x^{63} \cdot y^{-189}$

Step 3: Final expression with positive exponents

Since the problem asks for positive exponents, we can write $8^{-21}$ and $y^{-189}$ with positive exponents by moving them to the denominator:

$\frac{x^{63}}{8^{21} y^{189}}$

Thus, the simplified expression is:

$\frac{x^{63}}{8^{21} y^{189}}$

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Related Questions:

1. What is the rule for distributing exponents over multiplication inside parentheses?
2. How do you simplify negative exponents?
3. How can you express fractions using positive exponents?
4. What is the value of $8^{21}$?
5. How would the result change if the exponents inside the parentheses were positive initially?

Tip:

When simplifying expressions with exponents, always apply exponent rules step by step to avoid confusion and make sure to rewrite the final answer using positive exponents if asked.