Question

What is the cube root of 0.000001?

Answer 1

Hint: This type of question depends on the concept of cube-root of a number and properties of indices. The standard representation of a cube-root is \[\sqrt[3]{{}}\] which we can also express in terms of index as \[\sqrt[3]{x}={{x}^{\left( \dfrac{1}{3} \right)}}\]. Also here we have to use properties of indices such as, \[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}\], \[{{a}^{-m}}=\dfrac{1}{{{a}^{m}}}\].

Complete step by step solution:
Now, we have to find the cube-root of 0.000001
\[\Rightarrow 0.000001={{10}^{-6}}\]
By taking cube-root of both sides we get,
\[\Rightarrow \sqrt[3]{0.000001}=\sqrt[3]{{{10}^{-6}}}\]
By using, \[\sqrt[3]{x}={{x}^{\left( \dfrac{1}{3} \right)}}\] we can write,
\[\Rightarrow \sqrt[3]{0.000001}={{\left( {{10}^{-6}} \right)}^{\left( \dfrac{1}{3} \right)}}\]
Now, we will use \[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}\] so that we get,
\[\Rightarrow \sqrt[3]{0.000001}={{10}^{\left( -6\times \dfrac{1}{3} \right)}}\]
\[\Rightarrow \sqrt[3]{0.000001}={{10}^{-2}}\]
Finally, with the help of one more property of indices \[{{a}^{-m}}=\dfrac{1}{{{a}^{m}}}\] we can write,
\[\Rightarrow \sqrt[3]{0.000001}=\dfrac{1}{{{10}^{2}}}\]
\[\Rightarrow \sqrt[3]{0.000001}=\dfrac{1}{100}\]
\[\Rightarrow \sqrt[3]{0.000001}=0.01\]
Hence, the cube-root of 0.000001 is 0.01.

Note: In this type of question students may use different powers of 10 to represent 0.000001 and then solve this question in another way. One can find the cube-root of 0.000001 as follows:
\[\Rightarrow 0.000001=\dfrac{1}{1000000}\]
Taking cube-root of both sides we get,
\[\Rightarrow \sqrt[3]{0.000001}=\sqrt[3]{\dfrac{1}{1000000}}\]
\[\Rightarrow \sqrt[3]{0.000001}=\sqrt[3]{\dfrac{1}{{{10}^{6}}}}\]
By using, \[\sqrt[3]{x}={{x}^{\left( \dfrac{1}{3} \right)}}\] we can write,
\[\Rightarrow \sqrt[3]{0.000001}={{\left( \dfrac{1}{{{10}^{6}}} \right)}^{\left( \dfrac{1}{3} \right)}}\]
By using the property of indices, \[{{\left( \dfrac{a}{b} \right)}^{m}}=\dfrac{{{a}^{m}}}{{{b}^{m}}}\]
\[\Rightarrow \sqrt[3]{0.000001}=\left( \dfrac{{{1}^{\left( \dfrac{1}{3} \right)}}}{{{\left( {{10}^{6}} \right)}^{\left( \dfrac{1}{3} \right)}}} \right)\]
Now, we will use, \[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}\] in the denominator,
\[\Rightarrow \sqrt[3]{0.000001}=\dfrac{{{1}^{\left( \dfrac{1}{3} \right)}}}{\left( {{10}^{6\times \dfrac{1}{3}}} \right)}\]
\[\Rightarrow \sqrt[3]{0.000001}=\dfrac{{{1}^{\left( \dfrac{1}{3} \right)}}}{\left( {{10}^{2}} \right)}\]
As \[{{1}^{\left( \dfrac{1}{3} \right)}}=1\And {{10}^{2}}=100\] we can write,
\[\Rightarrow \sqrt[3]{0.000001}=\dfrac{1}{\left( 100 \right)}\]
\[\Rightarrow \sqrt[3]{0.000001}=0.01\]
Hence, the cube-root of 0.000001 is 0.01.
Some of the students may represent 0.000001 as \[\dfrac{1}{{{10}^{6}}}=\dfrac{1}{{{10}^{2}}\times {{10}^{2}}\times {{10}^{2}}}\]. Also, some of the students may represent 0.000001 as \[\dfrac{1}{{{10}^{6}}}=\dfrac{1}{{{10}^{3}}\times {{10}^{3}}}\]. And hence can find cube-root of 0.000001 in two different ways again as follows:
Case I
\[\begin{align}
  & \Rightarrow \sqrt[3]{0.000001}=\sqrt[3]{\dfrac{1}{{{10}^{6}}}} \\
 & \Rightarrow \sqrt[3]{0.000001}={{\left( \dfrac{1}{{{10}^{6}}} \right)}^{\left( \dfrac{1}{3} \right)}} \\
 & \Rightarrow \sqrt[3]{0.000001}={{\left( \dfrac{1}{{{10}^{3}}\times {{10}^{3}}} \right)}^{\left( \dfrac{1}{3} \right)}} \\
 & \Rightarrow \sqrt[3]{0.000001}=\left( \dfrac{{{1}^{^{\left( \dfrac{1}{3} \right)}}}}{{{\left( {{10}^{3}} \right)}^{^{\left( \dfrac{1}{3} \right)}}}\times {{\left( {{10}^{3}} \right)}^{^{\left( \dfrac{1}{3} \right)}}}} \right) \\
 & \Rightarrow \sqrt[3]{0.000001}=\dfrac{1}{10\times 10} \\
 & \Rightarrow \sqrt[3]{0.000001}=\dfrac{1}{100} \\
 & \Rightarrow \sqrt[3]{0.000001}=0.01 \\
\end{align}\]
Case II
Also, with the expression \[\dfrac{1}{{{10}^{6}}}=\dfrac{1}{{{10}^{2}}\times {{10}^{2}}\times {{10}^{2}}}\]some can solve as follows:
\[\begin{align}
  & \Rightarrow \sqrt[3]{0.000001}=\sqrt[3]{\dfrac{1}{{{10}^{6}}}} \\
 & \Rightarrow \sqrt[3]{0.000001}={{\left( \dfrac{1}{{{10}^{6}}} \right)}^{\left( \dfrac{1}{3} \right)}} \\
 & \Rightarrow \sqrt[3]{0.000001}={{\left( \dfrac{1}{{{10}^{2}}\times {{10}^{2}}\times {{10}^{2}}} \right)}^{\left( \dfrac{1}{3} \right)}} \\
 & \Rightarrow \sqrt[3]{0.000001}={{\left( \dfrac{1}{{{\left( {{10}^{2}} \right)}^{3}}} \right)}^{\left( \dfrac{1}{3} \right)}} \\
 & \Rightarrow \sqrt[3]{0.000001}=\dfrac{{{1}^{\left( \dfrac{1}{3} \right)}}}{{{\left( {{\left( {{10}^{2}} \right)}^{3}} \right)}^{\left( \dfrac{1}{3} \right)}}} \\
 & \Rightarrow \sqrt[3]{0.000001}=\dfrac{1}{{{10}^{2}}} \\
 & \Rightarrow \sqrt[3]{0.000001}=\dfrac{1}{100} \\
 & \Rightarrow \sqrt[3]{0.000001}=0.01 \\
\end{align}\]
Hence, the cube-root of 0.000001 is 0.01.