Multiply Square Roots: √2 × √3 × √1 × √4 × √2 Calculation

In order to simplify the given expression, apply two laws of exponents:

a. Root definition as an exponent:

$\sqrt[n]{a}=a^{\frac{1}{n}}$

b. The law of exponents for an exponent applied to a product in parentheses (in reverse direction):

$x^n\cdot y^n =(x\cdot y)^n$

Begin by converting the square roots to exponents using the law of exponents mentioned in a':

$\sqrt{2}\cdot\sqrt{3}\cdot\sqrt{1}\cdot\sqrt{4}\cdot\sqrt{2}= \\ \downarrow\\ 2^{\frac{1}{2}}\cdot3^{\frac{1}{2}}\cdot1^{\frac{1}{2}}\cdot4^{\frac{1}{2}}\cdot2^{\frac{1}{2}}=$

Due to the fact that there is a multiplication operation between five terms with identical exponents we are able to apply the law of exponents mentioned in b' (which of course also applies to multiplying several terms in parentheses) Combine them together in a multiplication operation inside of parentheses which are also raised to the same exponent:

$2^{\frac{1}{2}}\cdot3^{\frac{1}{2}}\cdot1^{\frac{1}{2}}\cdot4^{\frac{1}{2}}\cdot2^{\frac{1}{2}}= \\ (2\cdot3\cdot1\cdot4\cdot2)^{\frac{1}{2}}=\\ 48^{\frac{1}{2}}=\\ \boxed{\sqrt{48}}$

In the final steps, we first performed the multiplication inside of the parentheses, then we once again used the root definition as an exponent mentioned earlier in a' (in reverse direction) to return to root notation.

Therefore, we can identify that the correct answer is answer b.