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Explanation:

Given, ${\log }_{x^2}\left(y\right)$ + ${\log }_{y^2}\left(x\right)$ = 1

⇒ $\frac{{\log }_x\left(y\right)}{2}$ + $\frac{{\log }_y\left(x\right)}{2}$ = 1

⇒ log_xy + log_yx = 2

Now, log_xy and log_yx are reciprocal of each other and their sum is 2. Sum of a number and its reciprocal is always greater than or equal to 2. Equality is true when the number and reciprocal are both equal to 1.

∴ log_xy = 1
⇒ x = y

Now, y = x² - 30 [y = x]
⇒ x² - x - 30 = 0
⇒ (x - 6)(x + 5) = 0
⇒ x = 6 [x = -5 is rejected as log is not defined for negative numbers.]
∴ y = 6

∴ x² + y² = 6² + 6² = 72

Hence, 72.