To solve the equation \( x^2 = 7^8 \), we take the square root of both sides. This gives us:
\[ x = \pm \sqrt{7^8} \]
Recall that \( \sqrt{a^b} = a^{b/2} \), so we can simplify \( \sqrt{7^8} \):
\[ \sqrt{7^8} = 7^{8/2} = 7^4 \]
Thus, the solutions for \( x \) are:
\[ x = 7^4 \quad \text{and} \quad x = -7^4 \]
Calculating \( 7^4 \):
\[ 7^4 = 2401 \]
Therefore, the two real solutions are:
\[ -7^4 \quad \text{and} \quad 7^4 \]
The lesser solution is:
\[ -7^4 \]
So, the final answers are:
\[ -7^4, \quad 7^4 \]