Let's break down the given expression step by step. The expression is:

$\frac{7x^2y^{-3} \left( -2x^{-1}y^{-3} \right)^3}{(xy^5)^{-2}} - \frac{55y^{-2}}{x^{-1}}$

Step-by-Step Solution:

First Term Simplification:

$\frac{7x^2y^{-3} \left( -2x^{-1}y^{-3} \right)^3}{(xy^5)^{-2}}$

1. Simplify the power in the numerator:

$\left( -2x^{-1}y^{-3} \right)^3 = (-2)^3 \cdot (x^{-1})^3 \cdot (y^{-3})^3 = -8x^{-3}y^{-9}$

Now the first term becomes:

$\frac{7x^2y^{-3} \cdot (-8x^{-3}y^{-9})}{(xy^5)^{-2}}$

2. Multiply the terms in the numerator:

$7x^2y^{-3} \cdot (-8x^{-3}y^{-9}) = -56x^{2 + (-3)}y^{-3 + (-9)} = -56x^{-1}y^{-12}$

So, the expression is now:

$\frac{-56x^{-1}y^{-12}}{(xy^5)^{-2}}$

3. Simplify the denominator:

$(xy^5)^{-2} = x^{-2}y^{-10}$

Thus, the first term becomes:

$\frac{-56x^{-1}y^{-12}}{x^{-2}y^{-10}} = -56x^{-1 - (-2)}y^{-12 - (-10)} = -56x^{1}y^{-2}$

The simplified first term is:

$-56x^1y^{-2}$

Second Term Simplification:

The second term is:

$\frac{55y^{-2}}{x^{-1}}$

Simplify this:

$\frac{55y^{-2}}{x^{-1}} = 55x^1y^{-2}$

Putting Everything Together:

Now, the entire expression is:

$-56x^1y^{-2} - 55x^1y^{-2}$

Since the terms are like terms (both have $x^1y^{-2}$), we can combine them:

$(-56 - 55)x^1y^{-2} = -111x^1y^{-2}$

Final Answer:

$-111x y^{-2}$

Would you like more details on any of the steps or have any questions? Here are five related questions to consider:

1. How do you simplify expressions with negative exponents?
2. What are the properties of exponents when multiplying terms with the same base?
3. How can negative exponents be interpreted in terms of fractions?
4. What rules apply when raising powers of terms to another power?
5. How do you simplify complex fractions involving exponents?

Tip: When dealing with negative exponents, remember that $a^{-n} = \frac{1}{a^n}$, and applying this rule can make simplifications easier.