Question

What is the simplified value of \[{{\left( 81 \right)}^{\dfrac{-1}{4}}}+{{\left( 81 \right)}^{\dfrac{1}{4}}}\] ?

Answer 1

Hint: We are given an expression which we have to simplify and find the value of the expression. We can see that in the powers of the expression it has 4 so, we will remove them by writing 81 as the fourth power of 3, that is, \[{{3}^{4}}\]. We will then solve the expression further, and we will simplify the expression as much as possible and then we will have the value of the given expression.

Complete step-by-step answer:
According to the given question, we are given an expression which we have to simplify and find the value of the same.
The expression that we have is,
\[{{\left( 81 \right)}^{\dfrac{-1}{4}}}+{{\left( 81 \right)}^{\dfrac{1}{4}}}\]
We can see that the power has the number 4 with it, so in order to cancel that out, we will have to write the number 81 appropriately.
We can 81 as the fourth power of 3 and so the 4 in the powers gets cancelled and the expression gets simplified as well. So, we have,
\[= {{\left( {{3}^{4}} \right)}^{\dfrac{-1}{4}}}+{{\left( {{3}^{4}} \right)}^{\dfrac{1}{4}}}\]
Simplifying the powers now, we get,
\[= {{\left( 3 \right)}^{4\times }}^{\dfrac{-1}{4}}+{{\left( 3 \right)}^{4\times }}^{\dfrac{1}{4}}\]
We get the new expression as, we have,
\[= {{\left( 3 \right)}^{-1}}+{{\left( 3 \right)}^{1}}\]
The above expression can also be written as,
\[= \dfrac{1}{3}+3\]
We will now take the LCM and simplify the expression further and we get,
\[= \dfrac{1+9}{3}\]
Adding up the terms in the numerator, we get the expression as,
\[= \dfrac{10}{3}\]
Therefore, the value of the given expression is \[\dfrac{10}{3}\].

Note: The expression when in the process of simplification should be done carefully without missing any terms in the way, else the answer will be incorrect and in vain as well. So, the procedure should be clear and it should be done step wise. Also, while simplifying the powers, do not mix the terms of the powers and the normal together.