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.
Quantity I: Value of `((28)^4-(28)^3)/(42)^3`
Quantity II: Value of `3^2`

Options

A. Quantity `IgtQ` uantity II
B. Quantity `IltQ` uantity II
C. Quantity `IleQ` uantity II
D. Quantity `IgeQ` uantity II

Answer

To solve the problem, we will evaluate both Quantity I and Quantity II step by step.

### Step-by-Step Solution

**Step 1: Evaluate Quantity I**

We start with the expression given in Quantity I:

\[ \text{Quantity I} = \frac{(28)^4 - (28)^3}{(42)^3} \]

**Step 2: Factor the numerator**

We can factor out \( (28)^3 \) from the numerator:

\[ (28)^4 - (28)^3 = (28)^3 \cdot (28 - 1) = (28)^3 \cdot 27 \]

So, we can rewrite Quantity I as:

\[ \text{Quantity I} = \frac{(28)^3 \cdot 27}{(42)^3} \]

**Step 3: Simplify the denominator**

Next, we can express \( (42)^3 \) as \( 42 \cdot 42 \cdot 42 \).

**Step 4: Substitute values**

Now, we substitute the values into the expression:

\[ \text{Quantity I} = \frac{(28)^3 \cdot 27}{(42)^3} \]

**Step 5: Simplify the fraction**

To simplify this fraction, we can express \( 28 \) and \( 42 \) in terms of their prime factors:

\[ 28 = 4 \cdot 7 = 2^2 \cdot 7 \]
\[ 42 = 6 \cdot 7 = 2 \cdot 3 \cdot 7 \]

Thus,

\[ (28)^3 = (2^2 \cdot 7)^3 = 2^6 \cdot 7^3 \]
\[ (42)^3 = (2 \cdot 3 \cdot 7)^3 = 2^3 \cdot 3^3 \cdot 7^3 \]

Now we can substitute these back into Quantity I:

\[ \text{Quantity I} = \frac{2^6 \cdot 7^3 \cdot 27}{2^3 \cdot 3^3 \cdot 7^3} \]

**Step 6: Cancel out common terms**

The \( 7^3 \) cancels out:

\[ \text{Quantity I} = \frac{2^6 \cdot 27}{2^3 \cdot 3^3} \]

Now simplify \( 2^6 / 2^3 \):

\[ \text{Quantity I} = 2^{6-3} \cdot \frac{27}{3^3} = 2^3 \cdot \frac{27}{27} = 2^3 = 8 \]

**Step 7: Evaluate Quantity II**

Now we evaluate Quantity II:

\[ \text{Quantity II} = 3^2 = 9 \]

**Step 8: Compare the two quantities**

Now we compare Quantity I and Quantity II:

\[ \text{Quantity I} = 8 \]
\[ \text{Quantity II} = 9 \]

Since \( 8 < 9 \), we conclude that:

\[ \text{Quantity I} < \text{Quantity II} \]

### Final Answer

The correct relation is:

**Quantity I is less than Quantity II.**