Answer: The correct options are
(B) the value of b is ![\dfrac{1}{36}.](https://tex.z-dn.net/?f=%5Cdfrac%7B1%7D%7B36%7D.)
(C) As the value of the exponent decreases, each previous value is divided by 6.
Step-by-step explanation: We are given that Tori examined the pattern of exponents in the following table :
![\textup{power of 6}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\textup{value}\\\\6^3~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~216\\\\6^2~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~36\\\\6^1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~6\\\\6^0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~1\\\\6^{-1}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~a\\\\6^{-2}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~b](https://tex.z-dn.net/?f=%5Ctextup%7Bpower%20of%206%7D~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~%5Ctextup%7Bvalue%7D%5C%5C%5C%5C6%5E3~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~216%5C%5C%5C%5C6%5E2~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~36%5C%5C%5C%5C6%5E1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~6%5C%5C%5C%5C6%5E0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~1%5C%5C%5C%5C6%5E%7B-1%7D~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~a%5C%5C%5C%5C6%5E%7B-2%7D~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~b)
We are to select the true statements based on the above pattern.
We will be using the following property of exponents :
![x^{-y}=\dfrac{1}{x^y}.](https://tex.z-dn.net/?f=x%5E%7B-y%7D%3D%5Cdfrac%7B1%7D%7Bx%5Ey%7D.)
Therefore, we get
![a=6^{-1}=\dfrac{1}{6^1}=\dfrac{1}{6}.](https://tex.z-dn.net/?f=a%3D6%5E%7B-1%7D%3D%5Cdfrac%7B1%7D%7B6%5E1%7D%3D%5Cdfrac%7B1%7D%7B6%7D.)
and
![b=6^{-2}=\dfrac{1}{6^2}=\dfrac{1}{36}.](https://tex.z-dn.net/?f=b%3D6%5E%7B-2%7D%3D%5Cdfrac%7B1%7D%7B6%5E2%7D%3D%5Cdfrac%7B1%7D%7B36%7D.)
Also, the value of the exponent is decreasing and we see that
![\dfrac{216}{36}=\dfrac{36}{6}=\dfrac{6}{1}=\dfrac{1}{\frac{1}{6}}=\dfrac{\frac{1}{6}}{\frac{1}{36}}=6.](https://tex.z-dn.net/?f=%5Cdfrac%7B216%7D%7B36%7D%3D%5Cdfrac%7B36%7D%7B6%7D%3D%5Cdfrac%7B6%7D%7B1%7D%3D%5Cdfrac%7B1%7D%7B%5Cfrac%7B1%7D%7B6%7D%7D%3D%5Cdfrac%7B%5Cfrac%7B1%7D%7B6%7D%7D%7B%5Cfrac%7B1%7D%7B36%7D%7D%3D6.)
So, each previous value is divided by 6.
Thus, the correct options are
(B) the value of b is ![\dfrac{1}{36}.](https://tex.z-dn.net/?f=%5Cdfrac%7B1%7D%7B36%7D.)
(C) As the value of the exponent decreases, each previous value is divided by 6.