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

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 exponents applied to multiplication of terms 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:

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

Due to the fact that there is a multiplication operation between four terms with identical exponents, we are able to apply the law of exponents mentioned in b (which also applies to multiplication of multiple terms in parentheses) Combine them together in a multiplication operation within 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, we then once again used 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.