Determine the Largest Value Among Given Numbers

In order to determine which of the suggested options has the largest numerical value, apply the three laws of exponents:

a. Definition of root as an exponent:

$\sqrt[n]{a}=a^{\frac{1}{n}}$b. Law of exponents for an exponent applied to a product in parentheses (in reverse order):

$a^n\cdot b^n=(a\cdot b)^n$c. Law of exponents for an exponent raised to an exponent:

$(a^m)^n=a^{m\cdot n}$

Let's deal with each of the suggested options (in the answers), starting by converting the square root to exponent notation, using the law of exponents mentioned in a' earlier:

$$$\sqrt{5}\cdot\sqrt{5} \rightarrow 5^{\frac{1}{2}}\cdot5^{\frac{1}{2}}\\ \sqrt{2}\cdot\sqrt{2} \rightarrow 2^{\frac{1}{2}}\cdot2^{\frac{1}{2}}\\ \sqrt{3}\cdot\sqrt{3} \rightarrow 3^{\frac{1}{2}}\cdot3^{\frac{1}{2}}\\ \sqrt{4}\cdot\sqrt{4} \rightarrow 4^{\frac{1}{2}}\cdot4^{\frac{1}{2}}\\$Due to the fact that both terms in the product have the same exponent, we are able to apply the law of exponents mentioned in b' earlier and then proceed to combine them together inside of the parentheses product, raised to the same exponent . Once completed we can then calculate the result of the product in the parentheses:

$5^{\frac{1}{2}}\cdot5^{\frac{1}{2}} \rightarrow (5\cdot5)^{\frac{1}{2}}=(5^2)^{\frac{1}{2}} \\ 2^{\frac{1}{2}}\cdot2^{\frac{1}{2}}\rightarrow(2\cdot2)^{\frac{1}{2}}=(2^2)^{\frac{1}{2}} \\ 3^{\frac{1}{2}}\cdot3^{\frac{1}{2}} \rightarrow (3\cdot3)^{\frac{1}{2}}=(3^2)^{\frac{1}{2}} \\ 4^{\frac{1}{2}}\cdot4^{\frac{1}{2}}\rightarrow(4\cdot4)^{\frac{1}{2}}=(4^2)^{\frac{1}{2}} \\$Proceed to apply the law of exponents mentioned in c' and then calculate the exponent inside of the parentheses:

$(5^2)^{\frac{1}{2}}\rightarrow 5^{2\cdot \frac{1}{2}}=5^1=5 \\ (2^2)^{\frac{1}{2}}\rightarrow 2^{2\cdot \frac{1}{2}}=2^1=2 \\ (4^2)^{\frac{1}{2}}\rightarrow 4^{2\cdot \frac{1}{2}}=4^1=4 \\ (3^2)^{\frac{1}{2}}\rightarrow 3^{2\cdot \frac{1}{2}}=3^1=3 \\$

We have determined that the number in option a' is representative of the largest value:

$5>4>3>2$The correct answer is a'.