D01-01

Official Solution:


What is the value of \(\sqrt{324} + \sqrt{289}\)?


A. 31
B. 33
C. 34
D. 35
E. 36


The square root of a positive integer is either an integer or an irrational number. This means that the square root of a positive integer cannot be a fraction, such as \(\frac{1}{2}, \ \frac{3}{7}, \ \frac{19}{2}, \ \frac{1}{60}\), and so on. It must be either an integer (e.g., 1, 2, 3, ...) or an irrational number (e.g., \(\sqrt{2}\), \(2\sqrt{3}\), \(5\sqrt{7}\), ...). However, given that the answer choices provided are integers, both \(\sqrt{324}\) and \(\sqrt{289}\) must also be integers. This is because the sum of two positive irrational numbers, such as \(\sqrt{2}\), \(2\sqrt{3}\), \(5\sqrt{7}\), and so forth, will not result in an integer. This eliminates the possibility of irrational square roots in this context.

Next, \(\sqrt{324}\) is less than \(\sqrt{400}\) (which is 20), and must end with either 2 or 8. Given that \(12^2 = 144\), we can deduce that \(\sqrt{324} = 18\).

Consequently, \(\sqrt{289}\) must be less than 18 and end with either 3 or 7. Since \(13^2 = 169\), we can deduce that \(\sqrt{289} = 17\).

Therefore, the sum of \(\sqrt{324}\) and \(\sqrt{289}\) is \(18 + 17 = 35\).


Answer: D