evaluate \( a^{2}+\frac{1}{a^{2}} \) if \( a-\frac{1}{a}=7 \)

To evaluate \( a^{2}+\frac{1}{a^{2}} \) given that \( a-\frac{1}{a}=7 \), we start by squaring both sides of the equation: \[ \left(a - \frac{1}{a}\right)^{2} = 7^{2} \] This simplifies to: \[ a^{2} - 2 \cdot a \cdot \frac{1}{a} + \frac{1}{a^{2}} = 49 \] which simplifies further to: \[ a^{2} - 2 + \frac{1}{a^{2}} = 49 \] Now, we can rearrange this to find \( a^{2} + \frac{1}{a^{2}} \): \[ a^{2} + \frac{1}{a^{2}} = 49 + 2 = 51 \] Thus, the value of \( a^{2} + \frac{1}{a^{2}} \) is \( \boxed{51} \).