Multiply Square Roots: √5 × √10 × √2 × √4 Step-by-Step Solution

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

a. Root definition as an exponent:

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

b. The law of exponents for exponents applied to multiplication of terms in parentheses (in reverse order):

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

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

$\sqrt{5}\cdot\sqrt{10}\cdot\sqrt{2}\cdot\sqrt{4}= \\ \downarrow\\ 5^{\frac{1}{2}}\cdot10^{\frac{1}{2}}\cdot2^{\frac{1}{2}}\cdot4^{\frac{1}{2}}=$

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

$5^{\frac{1}{2}}\cdot10^{\frac{1}{2}}\cdot2^{\frac{1}{2}}\cdot4^{\frac{1}{2}}=\\ (5\cdot10\cdot2\cdot4)^{\frac{1}{2}}=\\ 400^{\frac{1}{2}}=\\ \sqrt{400}=\\ \boxed{20}$

In the final stages, we first performed the multiplication within the parentheses, then we used again the root definition as an exponent mentioned earlier in a (in reverse order) to return to root notation, and in the final stage we calculated the known square root of 400.

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