Evaluate : (ii) `(2a - 5b) ( 2a + 5b) ( 4a^2 + 25b^2)`

Answer

To evaluate the expression \((2a - 5b)(2a + 5b)(4a^2 + 25b^2)\), we can follow these steps:

### Step 1: Apply the difference of squares identity
We recognize that \((2a - 5b)(2a + 5b)\) can be simplified using the identity \( (x - y)(x + y) = x^2 - y^2 \).

Here, let \( x = 2a \) and \( y = 5b \). Therefore, we have:
\[ (2a - 5b)(2a + 5b) = (2a)^2 - (5b)^2 \]

### Step 2: Calculate the squares
Now we calculate the squares:
\[ (2a)^2 = 4a^2 \quad \text{and} \quad (5b)^2 = 25b^2 \]
Thus,
\[ (2a - 5b)(2a + 5b) = 4a^2 - 25b^2 \]

### Step 3: Substitute back into the expression
Now we substitute this result back into the original expression:
\[ (4a^2 - 25b^2)(4a^2 + 25b^2) \]

### Step 4: Apply the difference of squares identity again
We can again apply the difference of squares identity:
Let \( A = 4a^2 \) and \( B = 25b^2 \). Then:
\[ (4a^2 - 25b^2)(4a^2 + 25b^2) = A^2 - B^2 \]

### Step 5: Calculate \( A^2 \) and \( B^2 \)
Now we calculate:
\[ A^2 = (4a^2)^2 = 16a^4 \quad \text{and} \quad B^2 = (25b^2)^2 = 625b^4 \]
Thus,
\[ (4a^2 - 25b^2)(4a^2 + 25b^2) = 16a^4 - 625b^4 \]

### Final Answer
The final result of the evaluation is:
\[ 16a^4 - 625b^4 \]

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