If `y -(1)/(y) =4`, then find the value of `(y^(3)-(1)/(y^(3)))`

Options

A. A) 64
B. B) 76
C. C) 88
D. D) 90

Answer

To solve the equation \( y - \frac{1}{y} = 4 \) and find the value of \( y^3 - \frac{1}{y^3} \), we can follow these steps:

### Step 1: Start with the given equation
We have:
\[ y - \frac{1}{y} = 4 \]

### Step 2: Cube both sides
Cubing both sides gives us:
\[ \left( y - \frac{1}{y} \right)^3 = 4^3 \]
Calculating \( 4^3 \):
\[ 4^3 = 64 \]
So we have:
\[ \left( y - \frac{1}{y} \right)^3 = 64 \]

### Step 3: Use the identity for cubing a binomial
We can use the identity:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
In our case, let \( a = y \) and \( b = \frac{1}{y} \). Therefore:
\[ \left( y - \frac{1}{y} \right)^3 = y^3 - \frac{1}{y^3} - 3 \left( y \cdot \frac{1}{y} \right) \left( y - \frac{1}{y} \right) \]
This simplifies to:
\[ y^3 - \frac{1}{y^3} - 3(y - \frac{1}{y}) \]

### Step 4: Substitute known values
We know \( y - \frac{1}{y} = 4 \). Therefore:
\[ y^3 - \frac{1}{y^3} - 3(4) = 64 \]
This simplifies to:
\[ y^3 - \frac{1}{y^3} - 12 = 64 \]

### Step 5: Solve for \( y^3 - \frac{1}{y^3} \)
Now, we can isolate \( y^3 - \frac{1}{y^3} \):
\[ y^3 - \frac{1}{y^3} = 64 + 12 \]
Calculating this gives:
\[ y^3 - \frac{1}{y^3} = 76 \]

### Final Answer
Thus, the value of \( y^3 - \frac{1}{y^3} \) is:
\[ \boxed{76} \]