Consider the following statements :
`S_(1) : - 8 = 2ixx 4i = sqrt((-4)) xx sqrt((-16)) " " S_(2) : sqrt((-4) )xx sqrt((-16)) = sqrt( (-4) xx (-16))`
`S_(3) : sqrt((-4) xx (16) )= sqrt( -64)" " S_(4) : sqrt(64)= 8`
Of these statements, the incorrect one is :

Options

A. `S_(1)` only
B. `S_(2) ` only
C. `S_(3)` only
D. None of these

Answer

To solve the given problem, we need to evaluate the four statements provided and determine which one is incorrect. Let's analyze each statement step by step.

### Step 1: Evaluate Statement S1
**Statement S1:** \(-8 = 2i \cdot 4i = \sqrt{-4} \cdot \sqrt{-16}\)

1. Calculate \(2i \cdot 4i\):
\[ 2i \cdot 4i = 8i^2 = 8(-1) = -8 \]
So, the first part of S1 is correct.

2. Now, calculate \(\sqrt{-4} \cdot \sqrt{-16}\):
\[ \sqrt{-4} = 2i \quad \text{and} \quad \sqrt{-16} = 4i \]
Therefore,
\[ \sqrt{-4} \cdot \sqrt{-16} = 2i \cdot 4i = 8i^2 = -8 \]
Thus, the second part of S1 is also correct.

**Conclusion for S1:** S1 is correct.

### Step 2: Evaluate Statement S2
**Statement S2:** \(\sqrt{-4} \cdot \sqrt{-16} = \sqrt{-4 \cdot -16}\)

1. We already calculated \(\sqrt{-4} \cdot \sqrt{-16}\) in S1:
\[ \sqrt{-4} \cdot \sqrt{-16} = 2i \cdot 4i = -8 \]

2. Now, calculate \(\sqrt{-4 \cdot -16}\):
\[ -4 \cdot -16 = 64 \quad \text{so} \quad \sqrt{-4 \cdot -16} = \sqrt{64} = 8 \]

Since \(-8 \neq 8\), the statement is incorrect.

**Conclusion for S2:** S2 is incorrect.

### Step 3: Evaluate Statement S3
**Statement S3:** \(\sqrt{-4 \cdot 16} = \sqrt{-64}\)

1. Calculate \(-4 \cdot 16\):
\[ -4 \cdot 16 = -64 \]
Therefore,
\[ \sqrt{-4 \cdot 16} = \sqrt{-64} = 8i \]

**Conclusion for S3:** S3 is correct.

### Step 4: Evaluate Statement S4
**Statement S4:** \(\sqrt{64} = 8\)

1. Calculate \(\sqrt{64}\):
\[ \sqrt{64} = 8 \]

**Conclusion for S4:** S4 is correct.

### Final Conclusion
The only incorrect statement among S1, S2, S3, and S4 is **S2**.

### Summary of Steps
1. **S1**: Verified as correct.
2. **S2**: Found to be incorrect.
3. **S3**: Verified as correct.
4. **S4**: Verified as correct.

### Answer
The incorrect statement is **S2**.