Question

How do you simplify \[{{\left( \dfrac{1}{16} \right)}^{-\left( \dfrac{1}{4} \right)}}\]?

Answer 1

Hint: In this problem, we have to simplify the given expression, to its reduced form. To solve these types of problems, we should know exponent rules. In this problem we are going to use the negative property of exponent to simplify in a simple way and then we can simplify the fourth root to get the final answer.

Complete step by step answer:
We know that the given expression to be simplified is,
\[{{\left( \dfrac{1}{16} \right)}^{-\left( \dfrac{1}{4} \right)}}\] …… (1)
We also know that the Negative property of exponent is,
\[{{b}^{-n}}=\dfrac{1}{{{b}^{n}}}\]
Now we can apply the above negative property of exponent in the expression (1), we get
\[\begin{align}
  & \Rightarrow {{\left( \dfrac{1}{16} \right)}^{-\left( \dfrac{1}{4} \right)}}=\dfrac{1}{{{\left( \dfrac{1}{16} \right)}^{\dfrac{1}{4}}}} \\
 & \Rightarrow {{\left( \dfrac{16}{1} \right)}^{\dfrac{1}{4}}} \\
\end{align}\]
We know that one to the power anything is equal to one, so we can leave the denominator. Now, we can write the above step as,
\[\Rightarrow {{2}^{4\times \dfrac{1}{4}}}\text{ }\because {{\text{2}}^{4}}=16\]
Now, we can cancel the similar terms in the power, we get
\[\Rightarrow 2\]
Therefore, the simplified value of \[{{\left( \dfrac{1}{16} \right)}^{-\left( \dfrac{1}{4} \right)}}\] is 2.
Note: We also have another method to solve this problem,
We know that the given expression to be simplified is,
\[{{\left( \dfrac{1}{16} \right)}^{-\left( \dfrac{1}{4} \right)}}\] …… (1)
We also know that the Negative property of exponent is,
\[{{b}^{-n}}=\dfrac{1}{{{b}^{n}}}\]
Now we can apply the above negative property of exponent in the expression (1), we get
\[\begin{align}
  & \Rightarrow {{\left( \dfrac{1}{16} \right)}^{-\left( \dfrac{1}{4} \right)}}=\dfrac{1}{{{\left( \dfrac{1}{16} \right)}^{\dfrac{1}{4}}}} \\
 & \Rightarrow {{\left( \dfrac{16}{1} \right)}^{\dfrac{1}{4}}} \\
\end{align}\]
Now, we can take fourth root to get the value,
\[\begin{align}
  & \Rightarrow \sqrt[4]{16} \\
 & \Rightarrow \sqrt[4]{2\times 2\times 2\times 2} \\
 & \Rightarrow 2 \\
\end{align}\]
Therefore, the simplified value of \[{{\left( \dfrac{1}{16} \right)}^{-\left( \dfrac{1}{4} \right)}}\] is 2.

Note:
Students should know the exponent rules and properties to solve these types of problems, we can also use fourth root to solve these types of problems, when necessary.