Determine which of the following options has the greatest numerical value:
Video Solution
Solution Steps
00:00Select the largest value
00:03When multiplying the root of a number (A) by the root of another number (B)
00:06The result equals the root of their product (A times B)
00:09Apply this formula to our exercise and proceed to calculate the products
00:12We'll use this method for each expression in order to determine the largest one
00:25This is the solution
Step-by-Step Solution
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:
na=an1b. Law of exponents for an exponent applied to a product in parentheses (in reverse order):
an⋅bn=(a⋅b)nc. Law of exponents for an exponent raised to an exponent:
(am)n=am⋅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:
5⋅5→521⋅5212⋅2→221⋅2213⋅3→321⋅3214⋅4→421⋅421Due 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:
521⋅521→(5⋅5)21=(52)21221⋅221→(2⋅2)21=(22)21321⋅321→(3⋅3)21=(32)21421⋅421→(4⋅4)21=(42)21Proceed to apply the law of exponents mentioned in c' and then calculate the exponent inside of the parentheses: