Question

Evaluate the expression $ {\left( {0.01024} \right)^{ - \dfrac{2}{5}}} $ .

Answer 1

Hint: First make the prime factorization of the given number. Then use the rules of the indices in the simplification of the given expression and get the value of the expression given in the question.

Complete step-by-step answer:
Simplify the given expression by taking the fraction of the value given.
 $ {\left( {0.01024} \right)^{ - \dfrac{2}{5}}} = {\left( {\dfrac{{1024}}{{100000}}} \right)^{ - \dfrac{2}{5}}} $
Now make the numbers in the denominator and numerator as power of some number.
 $
  {\left( {0.01024} \right)^{ - \dfrac{2}{5}}} = {\left( {\dfrac{{1024}}{{100000}}} \right)^{ - \dfrac{2}{5}}} \\
   = {\left( {\dfrac{{{2^{10}}}}{{{{10}^5}}}} \right)^{ - \dfrac{2}{5}}} \;
  $
Now take the power separately for the numerator and denominator.
 $
  {\left( {0.01024} \right)^{ - \dfrac{2}{5}}} = {\left( {\dfrac{{{2^{10}}}}{{{{10}^5}}}} \right)^{ - \dfrac{2}{5}}} \\
   = \dfrac{{{{\left( {{2^{10}}} \right)}^{ - \dfrac{2}{5}}}}}{{{{\left( {{{10}^5}} \right)}^{ - \dfrac{2}{5}}}}} \;
  $
Now simplify the expression by the rule of indices $ {\left( {{a^b}} \right)^c} = {a^{bc}} $ .
 $
  {\left( {0.01024} \right)^{ - \dfrac{2}{5}}} = \dfrac{{{{\left( {{2^{10}}} \right)}^{ - \dfrac{2}{5}}}}}{{{{\left( {{{10}^5}} \right)}^{ - \dfrac{2}{5}}}}} \\
   = \dfrac{{{{\left( 2 \right)}^{10 \times - \dfrac{2}{5}}}}}{{{{\left( {10} \right)}^{5 \times - \dfrac{2}{5}}}}} \\
   = \dfrac{{{{\left( 2 \right)}^{ - 4}}}}{{{{\left( {10} \right)}^{ - 2}}}} \\
  $
Now solve further the expression by the rule of indices $ {a^{ - b}} = \dfrac{1}{{{a^b}}} $ .
 $
  {\left( {0.01024} \right)^{ - \dfrac{2}{5}}} = \dfrac{{{{\left( 2 \right)}^{ - 4}}}}{{{{\left( {10} \right)}^{ - 2}}}} \\
   = \dfrac{{\dfrac{1}{{{2^4}}}}}{{\dfrac{1}{{{{10}^2}}}}} \\
   = \dfrac{1}{{{2^4}}} \times \dfrac{{{{10}^2}}}{1} \\
   = \dfrac{{100}}{{16}} \\
   = \dfrac{{25}}{4} \\
   = 6.25 \;
  $
So, the value of $ {\left( {0.01024} \right)^{ - \dfrac{2}{5}}} $ is equal to $ 6.25 $ .
So, the correct answer is “ $ 6.25 $”.

Note: Make the given number as the power of any integer number and then use the rule of indices given below for the power and fractions powers also to simplify the value given in the question:
 $ {a^b}.{a^c} = {a^{b + c}} $ ,
 $ {a^{ - b}} = \dfrac{1}{{{a^b}}} $ and,
 $ {\left( {{a^b}} \right)^c} = {a^{bc}} $