Compute the following:
(i) $ {3^4} $
(ii) $ {\left( { - 5} \right)^2} $

Hint: We are given exponential number one has a base 3 and another has a base -5. 3 is a positive integer, so whatever the power it is raised to, it will result in another positive integer. -5 is a negative integer, so when -5 is raised to even powers the result will be positive and when it is raised to odd powers the result will be negative.

Complete step-by-step answer:
We are given to find the value of $ {3^4} $ and $ {\left( { - 5} \right)^2} $ .
I.First we are considering $ {3^4} $ . In $ {3^4} $ , the base is 3 and it is raised to the power 4. So the result consists of 3 multiplied 4 times with one another. As 3 is positive the result will also be positive.
Therefore, $ {3^4} $ is equal to $ 3 \times 3 \times 3 \times 3 = 81 $
So, the correct answer is “81”.

II.Next we have $ {\left( { - 5} \right)^2} $ . In $ {\left( { - 5} \right)^2} $ , the base is -5 and it is raised to the power 2. The result consists of -5 multiplied 2 times with one another. Here the exponent is 2 which is an even number. So the result will have a positive sign not a negative sign. If the exponent is an odd number for the base -5, then the result will have a negative sign.
Therefore, $ {\left( { - 5} \right)^2} $ is equal to $ \left( { - 5} \right) \times \left( { - 5} \right) = 25 $
So, the correct answer is “25”.

Note: A positive exponent means repeated multiplication of the base with the base, a negative exponent means repeated division of the base by the base. -5 will result negative when raised to odd powers because the sign of unlike integers is always negative and the sign of like integers is always positive.