If `a ^(2) + (1)/( a ^(2)) = 10,` then the value of `a ^(4) + (1)/( a ^(4)) ` is

Options

A. 90
B. 98
C. 200
D. 196

Answer

To solve the problem, we start with the given equation:

1. **Given Equation**:
\[ a^2 + \frac{1}{a^2} = 10 \]

2. **Square Both Sides**:
We will square both sides of the equation to find \( a^4 + \frac{1}{a^4} \).
\[ \left(a^2 + \frac{1}{a^2}\right)^2 = 10^2 \]

3. **Expand the Left Side**:
Using the identity \( (x + y)^2 = x^2 + y^2 + 2xy \), we expand the left side:
\[ a^4 + 2 + \frac{1}{a^4} = 100 \]

4. **Rearrange the Equation**:
Now, we can rearrange the equation to isolate \( a^4 + \frac{1}{a^4} \):
\[ a^4 + \frac{1}{a^4} = 100 - 2 \]

5. **Calculate the Final Value**:
Simplifying the right side gives us:
\[ a^4 + \frac{1}{a^4} = 98 \]

Thus, the value of \( a^4 + \frac{1}{a^4} \) is \( 98 \).