The powers 2 , 4 , 8 , 16 . . . of 2 grow very fast. Do the powers of 2 3 also increases like this? Justify your answer.

Given, the powers of 2 increases like 2 , 4 , 8 , 16 . . . Now, the given fraction is 2 3 Let's see, what happens to the fraction as the power increases. When the powers of 2 3 increases as 2 , 4 , 6 , 8 . . . , we get ⇒ 2 3 2 = 4 9 = 0 . 444 . . . . ⇒ 2 3 4 = 16 81 = 0 . 197530 ⇒ 2 3 6 = 64 729 = 0 . 0877 After comparing, we get 0 . 444 > 0 . 197530 > 0 . 0877 Here we can see that, the answer in each case is less than 1 and with every increase of power, the answer is decreasing. Hence, the powers of 2 3 will decrease.

The powers 2 , 4 , 8 , 16 . . . of 2 grow very fast. Do the powers of 3 2 also increases like this? Justify your answer

Given, the powers of 2 increases like 2 , 4 , 8 , 16 . . . Now, the given fraction is 3 2 Let's see, what happens to the fraction as the power increases. When the powers of 3 2 increases as 2 , 4 , 6 , 8 . . . , we get ⇒ 3 2 2 = 9 4 = 2 . 25 ⇒ 3 2 4 = 81 16 = 5 . 0625 ⇒ 3 2 6 = 729 64 = 11 . 390625 After comparing, we get 11 . 390625 > 5 . 0625 > 2 . 25 Here we can see that, the answer in each case with every increase of power, the answer is increasing. Hence, the powers of 3 2 will increase.