9.1. The square root of the number 49 can be extracted using this "formula": $\sqrt{49}=4+\sqrt{9}$. Are there other two-digit numbers whose square roots can be extracted in a similar manner and are integers? List all such two-digit numbers.
Answer: Yes, they exist: 64 and 81.

Solution. Consider all two-digit numbers that are squares of integers. The square roots of 16, 25, and 36 cannot be extracted in the specified manner, as the square roots of their last digits are not integers. The numbers 49, 64, and 81 are solutions.

The answer to the problem will not change if we do not require the root to be an integer. $\sqrt{10 \mathrm{a}+\mathrm{b}}=\mathrm{a}+\sqrt{\mathrm{b}}$, $10 \mathrm{a}+\mathrm{b}=\mathrm{a}^{2}+2 \mathrm{a} \sqrt{\mathrm{b}}+\mathrm{b}$. Since the left side of the equation is an integer, the number on the right side must also be an integer. Therefore, $\mathrm{b}=0,1,4$ or 9, which means $\mathrm{a}+\sqrt{\mathrm{b}}$ is an integer.