If ` y ^(4) + (1)/( y^(4)) = 223 and y gt 1 ` then find the value of ` ( y^(2) + (1) /( y^(2)))` ?

Options

A. 15
B. 14
C. 14.86
D. 16

Answer

To solve the equation \( y^4 + \frac{1}{y^4} = 223 \) and find the value of \( y^2 + \frac{1}{y^2} \), we can follow these steps:

### Step 1: Let \( x = y^2 + \frac{1}{y^2} \)

We start by letting \( x = y^2 + \frac{1}{y^2} \).

### Step 2: Relate \( y^4 + \frac{1}{y^4} \) to \( x \)

We know that:
\[ y^4 + \frac{1}{y^4} = \left( y^2 + \frac{1}{y^2} \right)^2 - 2 \]
This means:
\[ y^4 + \frac{1}{y^4} = x^2 - 2 \]

### Step 3: Substitute into the original equation

Now, we can substitute this into the original equation:
\[ x^2 - 2 = 223 \]

### Step 4: Solve for \( x^2 \)

Adding 2 to both sides gives us:
\[ x^2 = 223 + 2 = 225 \]

### Step 5: Solve for \( x \)

Taking the square root of both sides, we find:
\[ x = \sqrt{225} = 15 \]

### Conclusion

Thus, the value of \( y^2 + \frac{1}{y^2} \) is:
\[ \boxed{15} \]