M7MBA wrote:Which of the following is equivalent to √3+√2? $$\left(A\right)\ \sqrt{5}$$ $$\left(B\right)\ \sqrt{5+2\sqrt{6}}$$ $$\left(C\right)\ \sqrt{6+2\sqrt{5}}$$ $$\left(D\right)\ \sqrt{5-2\sqrt{6}}$$ $$\left(E\right)\ \sqrt{5+4\sqrt{6}}$$

Nice question!

KEY CONCEPT: If we take a positive number, square it and then take the square root of the result, we'll get back to the same number we started with.
For example...
Take: 5
Square it to get: 25
Take the square root: √25
Evaluate: 5

Now let's answer the question:
Take: √3+√2
Square it to get: (√3+√2)(√3+√2)
Use FOIL to expand : 3 + √6 + √6 + 2
Simplify: 5 + 2√6
Take the square root: √(5 + 2√6)

Answer: B

Cheers,
Brent

M7MBA wrote:Which of the following is equivalent to √3+√2? $$\left(A\right)\ \sqrt{5}$$ $$\left(B\right)\ \sqrt{5+2\sqrt{6}}$$ $$\left(C\right)\ \sqrt{6+2\sqrt{5}}$$ $$\left(D\right)\ \sqrt{5-2\sqrt{6}}$$ $$\left(E\right)\ \sqrt{5+4\sqrt{6}}$$ The Oa is the option B.

Experts, how can I rewrite the given expression to get one of the listed options? Can you show me the calculation needed? <i class="em em-confused"></i>

We see that the given choices are all in square roots. So if we square (√3 + √2), it should give us the radicand in one of the given choices.

(√3 + √2)^2 = (√3)^2 + (√2)^2 + 2(√3)(√2) = 3 + 2 + 2√6 = 5 + 2√6

Since (√3 + √2)^2 = 5 + 2√6, then √(5 + 2√6) = √3 + √2.

Answer: B