In the following question two Quantities i.e., Quantity I and Quantity II are given. You have to determine the relation between Quantity I and Quantity II.
Given, a, b, c and d are positive integers.
I : `a^(-b)/a^(-a)=a^b xx c`
II : `(a^3xxb^3)/(axxb^2)=(b^3xxd^4)/(db)`
Quantity I: Value of ‘c’
Quantity II: Value of ‘d’

Options

A. Quantity I `gt` Quantity II
B. Quantity I `lt` Quantity II
C. Quantity I `ge` Quantity II
D. Quantity I = Quantity II or no relation can be established

Explanation

To solve the given problem, we will analyze both quantities step by step.

### Step 1: Solve for Quantity I (Value of 'c')

The equation given is:
\[ \frac{a^{-b}}{a^{-a}} = a^b \cdot c \]

Using the property of exponents, we can simplify the left side:
\[ \frac{a^{-b}}{a^{-a}} = a^{-b - (-a)} = a^{-b + a} = a^{a - b} \]

Now, we equate this to the right side:
\[ a^{a - b} = a^b \cdot c \]

To isolate \(c\), we can divide both sides by \(a^b\):
\[ c = \frac{a^{a - b}}{a^b} = a^{(a - b) - b} = a^{a - 2b} \]

Thus, we have:
\[ c = a^{a - 2b} \]

### Step 2: Solve for Quantity II (Value of 'd')

The equation given is:
\[ \frac{a^3 \cdot b^3}{a \cdot b^2} = \frac{b^3 \cdot d^4}{b \cdot d} \]

Simplifying the left side:
\[ \frac{a^3 \cdot b^3}{a \cdot b^2} = \frac{a^3}{a} \cdot \frac{b^3}{b^2} = a^{3-1} \cdot b^{3-2} = a^2 \cdot b \]

Now simplifying the right side:
\[ \frac{b^3 \cdot d^4}{b \cdot d} = \frac{b^3}{b} \cdot \frac{d^4}{d} = b^{3-1} \cdot d^{4-1} = b^2 \cdot d^3 \]

Now we have:
\[ a^2 \cdot b = b^2 \cdot d^3 \]

To isolate \(d\), we can rearrange:
\[ d^3 = \frac{a^2 \cdot b}{b^2} = \frac{a^2}{b} \]

Taking the cube root of both sides gives:
\[ d = \sqrt[3]{\frac{a^2}{b}} \]

### Step 3: Compare Quantity I and Quantity II

Now we have:
- Quantity I: \(c = a^{a - 2b}\)
- Quantity II: \(d = \sqrt[3]{\frac{a^2}{b}}\)

### Step 4: Determine the relationship between Quantity I and Quantity II

Since \(a\), \(b\), \(c\), and \(d\) are all positive integers, we cannot definitively state a relationship between \(c\) and \(d\) without knowing specific values of \(a\) and \(b\). Therefore, we conclude that there is no definitive relationship between Quantity I and Quantity II.

### Final Answer:
There is no relation among them.

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