Question

How do u solve \[{{\left( \dfrac{1}{1000} \right)}^{-\dfrac{1}{3}}}\]?

Answer 1

Hint: In this problem, we are going to reduce the fraction into its simplest form. Here we have to solve the fraction part and also its power part which is also in the fraction form. Here we are going to use simple steps to reduce the fraction and to solve the problem. These types of problems can be solved efficiently when basic simplification is done. In this problem, in the numerator, anything to the power one is one itself, so we can assume the numerator as one and we should solve the denominator to get the solution.

Complete step by step answer:
We know that the given fraction is,
\[{{\left( \dfrac{1}{1000} \right)}^{-\dfrac{1}{3}}}\]
Now we can apply the whole power to its numerator and its denominator.
We get,
\[\begin{align}
  & \Rightarrow {{\left( \dfrac{1}{1000} \right)}^{-\dfrac{1}{3}}} \\
 & \Rightarrow \left( \dfrac{{{\left( 1 \right)}^{-\dfrac{1}{3}}}}{{{\left( 1000 \right)}^{-\dfrac{1}{3}}}} \right) \\
\end{align}\]
We know that anything to the power one is one itself, so the numerator becomes 1.
Now the fraction becomes,
\[\begin{align}
  & \Rightarrow \left( \dfrac{1}{{{\left( 1000 \right)}^{-\dfrac{1}{3}}}} \right) \\
 & \\
\end{align}\]
Now we have to solve the denominator,
On solving the denominator, we get
\[\begin{align}
  & \Rightarrow \left( \dfrac{1}{{{\left( 1000 \right)}^{-\dfrac{1}{3}}}} \right) \\
 & \Rightarrow {{\left( 1000 \right)}^{\dfrac{1}{3}}} \\
\end{align}\]
Here we can see that, by reciprocal, the negative power gets converted into the positive power.
Now by changing the fraction power to its cubic root form, we get
\[\begin{align}
  & \Rightarrow \sqrt[3]{1000} \\
 & \Rightarrow \sqrt[3]{10\times 10\times 10} \\
 & \Rightarrow 10 \\
\end{align}\]
Therefore, the solution of \[{{\left( \dfrac{1}{1000} \right)}^{-\dfrac{1}{3}}}\]is 10.

Note:
We also have another method to solve this problem. We know that the given fraction is,
\[{{\left( \dfrac{1}{1000} \right)}^{-\dfrac{1}{3}}}\]
Here we can take reciprocal of the fraction part, we get
\[\begin{align}
  & \Rightarrow {{\left( {{\left( \dfrac{1000}{1} \right)}^{-1}} \right)}^{-\dfrac{1}{3}}} \\
 & \Rightarrow {{\left( 1000 \right)}^{-1\times \dfrac{-1}{3}}} \\
 & \Rightarrow {{\left( 1000 \right)}^{\dfrac{1}{3}}} \\
 & \\
\end{align}\]
Now we can simplify it to get the answer,
\[\begin{align}
  & \Rightarrow {{\left( 10 \right)}^{3\times \dfrac{1}{3}}} \\
 & \Rightarrow 10 \\
\end{align}\]
Therefore, the solution of \[{{\left( \dfrac{1}{1000} \right)}^{-\dfrac{1}{3}}}\]is 10.
To solve these types of problems, students should know basic simplification steps to be followed in the fraction parts and also the reciprocal methods in case of negative fraction powers.