If a curve is represented parametrically by the equations `x=4t^(3)+3` and `y=4+3t^(4)` and `(d^(2)x)/(dy^(2))/((dx)/(dy))^(n)` is constant then the value of n, is

Answer

To solve the problem, we need to find the value of \( n \) such that the expression

\[ \frac{d^2x}{dy^2} \left( \frac{dx}{dy} \right)^n \]

is constant. We start with the parametric equations given:

\[ x = 4t^3 + 3 \]
\[ y = 4 + 3t^4 \]

### Step 1: Find \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \)

First, we differentiate \( x \) and \( y \) with respect to \( t \):

\[ \frac{dx}{dt} = \frac{d}{dt}(4t^3 + 3) = 12t^2 \]

\[ \frac{dy}{dt} = \frac{d}{dt}(4 + 3t^4) = 12t^3 \]

### Step 2: Find \( \frac{dy}{dx} \)

Next, we find \( \frac{dy}{dx} \) using the chain rule:

\[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{12t^3}{12t^2} = t \]

### Step 3: Find \( \frac{dx}{dy} \)

The reciprocal gives us \( \frac{dx}{dy} \):

\[ \frac{dx}{dy} = \frac{1}{t} \]

### Step 4: Find \( \frac{dt}{dy} \)

Now, we need to find \( \frac{dt}{dy} \):

Using the relationship \( \frac{dy}{dt} \):

\[ \frac{dt}{dy} = \frac{1}{\frac{dy}{dt}} = \frac{1}{12t^3} \]

### Step 5: Find \( \frac{d^2x}{dy^2} \)

To find \( \frac{d^2x}{dy^2} \), we can use the chain rule again:

\[ \frac{d^2x}{dy^2} = \frac{d}{dy} \left( \frac{dx}{dy} \right) = \frac{d}{dt} \left( \frac{dx}{dy} \right) \cdot \frac{dt}{dy} \]

Calculating \( \frac{d}{dt} \left( \frac{dx}{dy} \right) \):

\[ \frac{d}{dt} \left( \frac{dx}{dy} \right) = \frac{d}{dt} \left( \frac{1}{t} \right) = -\frac{1}{t^2} \]

Now substituting \( \frac{dt}{dy} \):

\[ \frac{d^2x}{dy^2} = -\frac{1}{t^2} \cdot \frac{1}{12t^3} = -\frac{1}{12t^5} \]

### Step 6: Substitute into the expression

Now we substitute \( \frac{d^2x}{dy^2} \) and \( \frac{dx}{dy} \) into the expression:

\[ \frac{d^2x}{dy^2} \left( \frac{dx}{dy} \right)^n = -\frac{1}{12t^5} \cdot (t)^n = -\frac{1}{12} t^{n-5} \]

### Step 7: Set the expression to be constant

For the expression to be constant, the exponent of \( t \) must be zero:

\[ n - 5 = 0 \implies n = 5 \]

### Final Answer

Thus, the value of \( n \) is

\[ \boxed{5} \]