Question

Find the square root of $$12+2\sqrt{35}$$.

Answer 1

Hint: In such types of questions, try to convert the given number into a perfect square. This enables us to cancel the root and the square so that we can get the final answer. The formula used will be-

Complete step-by-step answer:
a2 + b2 + 2ab = (a + b)2

We have to convert the given number into the form a2 + b2 + 2ab.

When we look closely, 5 + 7 = 12 and $$\sqrt{5\times7}\;=\sqrt{35}$$, So when we take

$$\mathrm a=\sqrt7\;\mathrm{and}\;\mathrm b=\sqrt5$$

Then we can write it as-

$$=\left(\sqrt7\right)^2+\left(\sqrt5\right)^2+2\times\sqrt7\times\sqrt5\\=\left(\sqrt7+\sqrt5\right)^2$$

The square root of this number can now be easily found-

$$=\sqrt{\left(\sqrt7+\sqrt5\right)^2}\\=\sqrt7+\sqrt5$$

This is the required answer.

Note: Students may write two solutions to these types of questions. But we should keep in mind that the square root of a number cannot be negative, so we write only the positive root.