If x=97+56\sqrt{3}, then what is the value of \sqrt{x}+\frac{1}{\sqrt{x}}?

Explanation

Observe that 97+56\sqrt{3} = (7+4\sqrt{3})^2. Taking the square root gives \sqrt{x} = 7+4\sqrt{3}, which further equals (2+\sqrt{3})^2. Thus, \sqrt{x} = 2+\sqrt{3}. Its reciprocal is \frac{1}{2+\sqrt{3}} = 2-\sqrt{3}. Their sum is 2+\sqrt{3} + 2-\sqrt{3} = 4.

Related questions on Arithmetic

• What is the remainder when (17^{25} + 19^{25}) is divided by 18?
• A bottle contains spirit and water in the ratio 1:4 and another identical bottle contains spirit and water in the ratio 4:1. In what rat...
• Let P = 5^5 \times 15^{15} \times 25^{25} \times 35^{35} and Q = 10^{10} \times 20^{20} \times 30^{30} \times 40^{40}. What is the numbe...
• Two students X and Y appeared in a test. The score of X is 20 more than that of Y. If the score of X is 75% of the sum of the scores of X an...
• Question: The product of a natural number N and the number M written by the same digits of N in the reverse order is 252. What is the number...