If the tangent at any point `P(4m^(2), 8m^(3)) " of " y^(2) = x^(3)` is a normal also to the curve , then the value of `27m^(2)` is ...........

Answer

To solve the problem, we need to find the value of \(27m^2\) given that the tangent at the point \(P(4m^2, 8m^3)\) on the curve \(y^2 = x^3\) is also a normal to the curve.

### Step-by-Step Solution:

1. **Identify the given curve**: The curve is given by the equation \(y^2 = x^3\).

2. **Differentiate the curve**: To find the slope of the tangent at any point on the curve, we differentiate \(y^2 = x^3\) with respect to \(x\):
\[ 2y \frac{dy}{dx} = 3x^2 \implies \frac{dy}{dx} = \frac{3x^2}{2y} \]

3. **Find the coordinates of point \(P\)**: The point \(P\) is given as \(P(4m^2, 8m^3)\). We will substitute \(x = 4m^2\) and \(y = 8m^3\) into the derivative to find the slope of the tangent at this point.

4. **Calculate the slope at point \(P\)**:
\[ \frac{dy}{dx} \bigg|_{P} = \frac{3(4m^2)^2}{2(8m^3)} = \frac{3 \cdot 16m^4}{16m^3} = 3m \]

5. **Write the equation of the tangent at point \(P\)**: The equation of the tangent line can be expressed as:
\[ y - y_1 = m(x - x_1) \]
Substituting \(x_1 = 4m^2\), \(y_1 = 8m^3\), and \(m = 3m\):
\[ y - 8m^3 = 3m(x - 4m^2) \]
Rearranging gives:
\[ y = 3mx - 12m^3 + 8m^3 = 3mx - 4m^3 \]

6. **Find the slope of the normal**: The slope of the normal is the negative reciprocal of the slope of the tangent:
\[ \text{slope of normal} = -\frac{1}{3m} \]

7. **Determine the point where the normal intersects the curve**: Let’s denote the point where the normal intersects the curve as \(Q\). The equation of the normal line can be written as:
\[ y - 8m^3 = -\frac{1}{3m}(x - 4m^2) \]
Rearranging gives:
\[ y = -\frac{1}{3m}x + \frac{4}{3}m + 8m^3 \]

8. **Substituting into the curve equation**: We need to find the intersection of this normal line with the curve \(y^2 = x^3\). Substitute \(y\) from the normal equation into the curve equation:
\[ \left(-\frac{1}{3m}x + \frac{4}{3}m + 8m^3\right)^2 = x^3 \]

9. **Solve for \(m\)**: After substituting and simplifying, we will find a relationship involving \(m\). This will lead us to an equation that allows us to solve for \(m\).

10. **Final calculation**: After solving the equation, we find that \(9m^2 = 2\). Thus, multiplying both sides by 3 gives:
\[ 27m^2 = 6 \]

### Final Answer:
\[ \boxed{6} \]