Select the largest value from among the given options:
Select the largest value from among the given options:
In order to determine which of the suggested options has the largest numerical value, we will apply two laws of exponents:
a. Definition of root as an exponent:
b. Law of exponents for exponents in parentheses (in reverse direction):
Let's proceed to examine the options a and c (in the answers), starting by converting the square root to exponent notation, using the law of exponents mentioned in a earlier:
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 to combine them inside of parentheses, which are subsequently raised to the same exponent. Once completed proceed to calculate the result of the multiplication inside of the parentheses:
In the next step, we will return to root notation, again, using the law of exponents mentioned in a (in reverse direction):
We can deduce that the numerical values of options a, b, and c are equal, as seen below:
Therefore, we need to determine which of these expressions:
has a higher numerical value,
This can be achieved by converting these two values to exponent notation, again, using the law of exponents mentioned in a:
Note that these two expressions have the same exponent (and their bases are positive), Therefore we can determine their relationship by simply comparing their bases, since it will be identical:
9>6\hspace{4pt} (>0)\\ \downarrow\\ 9^{\frac{1}{2}}>6^{\frac{1}{2}} In other words, we got that:
\sqrt{9}>\sqrt{6}
Therefore, the correct answer is answer d.