Multiply Square Roots: Solving √2·√2·√2·√1·√1 Step-by-Step

In order to simplify the given expression, we will use two laws of exponents:

a. The definition of root 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):

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

Let's start by converting the square roots to exponents using the law of exponents mentioned in a':

$\sqrt{2}\cdot\sqrt{2}\cdot\sqrt{2}\cdot\sqrt{1}\cdot\sqrt{1}= \\ \downarrow\\ 2^{\frac{1}{2}}\cdot2^{\frac{1}{2}}\cdot2^{\frac{1}{2}}\cdot1^{\frac{1}{2}}\cdot1^{\frac{1}{2}}=$

We'll continue, since there is a multiplication between five terms with identical exponents we can use the law of exponents mentioned in b' (which of course also applies to multiplying several terms in parentheses) and combine them together in a multiplication under parentheses which are raised to the same exponent:

$2^{\frac{1}{2}}\cdot2^{\frac{1}{2}}\cdot2^{\frac{1}{2}}\cdot1^{\frac{1}{2}}\cdot1^{\frac{1}{2}}= \\ (2\cdot2\cdot2\cdot1\cdot1)^{\frac{1}{2}}=\\ 8^{\frac{1}{2}}=\\ \boxed{\sqrt{8}}$

In the final steps, we first performed the multiplication within the parentheses, then we used again the definition of root 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 a'.