1

Problem

Simplify the expression \$\sqrt(5 + 2\sqrt6) + \sqrt(5 - 2\sqrt6)\$.

2

Step

The given expression is

\$\sqrt(5 + 2\sqrt6) + \sqrt(5 - 2\sqrt6)\$

3

Step

Start with the expression:

\$\sqrt(5 + 2\sqrt6) + \sqrt(5 - 2\sqrt6)\$

4

Clue

First, let’s simplify the individual square roots within each radical term:
\$ \sqrt(a+b) = \sqrt(a) + \sqrt(b) \$

\$ \sqrt(a-b) = \sqrt(a) - \sqrt(b) \$

5

Step

Now let’s apply these simplification rules to our expression:

\$\sqrt(5 + 2\sqrt6) + \sqrt(5 - 2\sqrt6)\$ = \$ \sqrt(\sqrt(5)^2 + \sqrt(2\sqrt(6))^2) \$ + \$ \sqrt(\sqrt(5)^2 - \sqrt(2\sqrt(6))^2) \$

6

Step

Make it simplify:

\$\sqrt(5 + 2\sqrt6) + \sqrt(5 - 2\sqrt6)\$ = \$ \sqrt5 + \cancel(2\sqrt(6)) + \sqrt5 - \cancel(2\sqrt(6) )\$

7

Step

After simplification:

\$\sqrt(5 + 2\sqrt6) + \sqrt(5 - 2\sqrt6)\$ = \$ \sqrt5 + \sqrt5 \$

\$\sqrt(5 + 2\sqrt6) + \sqrt(5 - 2\sqrt6)\$ = \$ 2\sqrt5 \$

\$ 2\sqrt5 \$

8

Solution

Therefore, the simplified expression is \$ 2\sqrt5 \$.

9

Sumup

Please summarize steps

Choices

10

Choice-A

\$10\sqrt(5)\$, is incorrect because it’s not possible to simplify the expression to \$\sqrt(5)\$ without any other term

Wrong \$ 10\sqrt5 \$

11

Choice-B

Incorrect: It incorrectly assumes that both terms are \$\sqrt(6)\$, which is not the case

Wrong \$ 3\sqrt6 \$

12

Choice-C

Correctly calculated using hint and clue, it was done with precision and accuracy

Correct \$ 2\sqrt5 \$

13

Choice-D

Wrong: Because the expression does not yield a negative result

Wrong \$ -2\sqrt5 \$

14

Answer

Option

C

15

Sumup

Please summarize choices