Numerical Comparison: Identify the Maximum Value Among Given Numbers

In order to determine which of the following options has the largest numerical value, we will apply two laws of exponents:

a. Definition of root as an exponent:

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

b. Law of exponents for exponents in parentheses (in reverse order):

$a^n\cdot b^n=(a\cdot b)^n$

Let's start by converting the fourth root in each of the suggested options to exponent notation, using the law of exponents mentioned in a above:

$$$\sqrt{2}\cdot\sqrt{1} \rightarrow 2^{\frac{1}{2}}\cdot1^{\frac{1}{2}}\\ \sqrt{2}\cdot\sqrt{2} \rightarrow 2^{\frac{1}{2}}\cdot2^{\frac{1}{2}}\\ \sqrt{2}\cdot\sqrt{3} \rightarrow 2^{\frac{1}{2}}\cdot3^{\frac{1}{2}}\\ \sqrt{2}\cdot\sqrt{4} \rightarrow 2^{\frac{1}{2}}\cdot4^{\frac{1}{2}}\\$Due to the fact that both terms in the multiplication have the same exponent, we are able to apply the law of exponents mentioned in b above and combine them together in the multiplication within parentheses, whilst raised to the same exponent. Once completed we can then calculate the result of the multiplication inside of the parentheses:

$2^{\frac{1}{2}}\cdot2^{\frac{1}{2}} \rightarrow (2\cdot1)^{\frac{1}{2}}=2^{\frac{1}{2}} \\ 2^{\frac{1}{2}}\cdot2^{\frac{1}{2}}\rightarrow(2\cdot2)^{\frac{1}{2}}=4^{\frac{1}{2}} \\ 2^{\frac{1}{2}}\cdot3^{\frac{1}{2}} \rightarrow (2\cdot3)^{\frac{1}{2}}=6^{\frac{1}{2}} \\ 2^{\frac{1}{2}}\cdot4^{\frac{1}{2}}\rightarrow(2\cdot4)^{\frac{1}{2}}=8^{\frac{1}{2}} \\$Let's summarize what we've done so far, as shown below:

$\sqrt{2}\cdot\sqrt{1}=2^{\frac{1}{2}}\\ \sqrt{2}\cdot\sqrt{2}= 4^{\frac{1}{2}}\\ \sqrt{2}\cdot\sqrt{3}= 6^{\frac{1}{2}}\\ \sqrt{2}\cdot\sqrt{4}= 8^{\frac{1}{2}}\\$Now let's note that all the expressions we obtained have the same exponent (they're bases are also positive), therefore we can determine the trend between them using only the trend between their bases, since it's identical to it:

$8>6>4>2\hspace{4pt} (>0)\\ \downarrow\\ 8^{\frac{1}{2}}>6^{\frac{1}{2}} >4^{\frac{1}{2}}>2^{\frac{1}{2}}$

Therefore the correct answer is answer d.