Multiply Square Roots: √5 × √2 × √5 × √2 Simplification

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 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:

$\sqrt{5}\cdot\sqrt{2}\cdot\sqrt{5}\cdot\sqrt{2}= \\ \downarrow\\ 5^{\frac{1}{2}}\cdot2^{\frac{1}{2}}\cdot5^{\frac{1}{2}}\cdot2^{\frac{1}{2}}=$

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 multiple terms in parentheses) and combine them together in multiplication under parentheses that are raised to the same exponent:

$5^{\frac{1}{2}}\cdot2^{\frac{1}{2}}\cdot5^{\frac{1}{2}}\cdot2^{\frac{1}{2}}= \\ (5\cdot2\cdot5\cdot2)^{\frac{1}{2}}=\\ 100^{\frac{1}{2}}=\\ \sqrt{100}=\\ \boxed{10}$

In the final steps, we first performed the multiplication within the parentheses, then we used again the root definition as an exponent mentioned in a (in reverse direction) to return to root notation, and in the final stage, we calculated the known square root of 100.

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