Determine the Largest Value Among Given Numbers

To determine which of the suggested options has the largest numerical value, we will use 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}}\\$Let's continue, since both terms in the product (in both options we are currently dealing with) have the same exponent, we can use the law of exponents mentioned in b' earlier and combine them in the parentheses product raised to the same exponent and then calculate the result of the product in parentheses:

$5^{\frac{1}{2}}\cdot5^{\frac{1}{2}} \rightarrow (5\cdot5)^{\frac{1}{2}}=(5^2)^{\frac{1}{2}} \\ 2^{\frac{1}{2}}\cdot2^{\frac{1}{2}}\rightarrow(2\cdot2)^{\frac{1}{2}}=(2^2)^{\frac{1}{2}} \\ 3^{\frac{1}{2}}\cdot3^{\frac{1}{2}} \rightarrow (3\cdot3)^{\frac{1}{2}}=(3^2)^{\frac{1}{2}} \\ 4^{\frac{1}{2}}\cdot4^{\frac{1}{2}}\rightarrow(4\cdot4)^{\frac{1}{2}}=(4^2)^{\frac{1}{2}} \\$Let's continue, we'll apply the law of exponents mentioned in c' and calculate the exponent on the term (with exponent) in 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 therefore found that the number in option a' is the largest because:

5>4>3>2 Therefore, the correct answer is answer a'.