Question

Calculate: ${\left( {1001} \right)^{1/3}}$

Answer 1

Hint: Here we have to calculate the cube root of a big number, so we will use the concept of binomial expansion of exponents and their laws to evaluate the cube root of 1001.

Complete step by step solution:
Now, we have to find the value of ${\left( {1001} \right)^{1/3}}$. To do that, first we will simplify and open the brackets and rewrite it as:
${\left( {1001} \right)^{1/3}} = {(1000 + 1)^{1/3}}$
Which can be further transformed by taking 1000 common as:
$ = {\left[ {1000\left( {1 + \dfrac{1}{{1000}}} \right)} \right]^{1/3}}$
Now solving further by taking the cube roots on the right hand side:
$
   = {\left[ {{{\left( {10} \right)}^3}\left( {{{\left( 1 \right)}^3} + \dfrac{1}{{{{\left( {10} \right)}^3}}}} \right)} \right]^{1/3}} \\
   = {\left[ {10\left( {1 + 0.001} \right)} \right]^{1/3}} \\
 $
Now, from binomial theorem we know that the expansion of ${(1 + x)^n} = 1 + nx + \dfrac{{n(n - 2}}{{2!}}{x^2}$
So, according to the question, x = 0.001 and n = 1/3.
Now using the formula for expansion as per Binomial Theorem for ${\left( {1 + 0.001} \right)^{1/3}}$ we will simplify the expression and get:

$
  {\left( {1 + 0.001} \right)^{1/3}} \\
   = \left( {1 + \dfrac{1}{3} \times 0.001 + \dfrac{{\dfrac{1}{3}\left( {\dfrac{1}{3} - 1} \right)}}{{2!}}{{\left( {0.001} \right)}^2}} \right) \\
   = \left( {1 + 0.0003333 + \dfrac{{\dfrac{1}{3}\left( {\dfrac{{1 - 3}}{3}} \right)}}{{2!}}{{\left( {0.001} \right)}^2}} \right) \\
   = \left( {1 + 0.0003333 + \dfrac{{\dfrac{1}{3}\left( { - \dfrac{2}{3}} \right)}}{{2 \times 1}}{{\left( {0.001} \right)}^2}} \right) \\
   = \left( {1 + 0.0003333 + \dfrac{{ - 2}}{{9 \times 2 \times 1}}{{\left( {0.001} \right)}^2}} \right) \\
   = \left( {1 + 0.0003333 - \dfrac{1}{9}{{\left( {0.001} \right)}^2}} \right) \\
   = \left( {1 + 0.0003333 - 0.000001} \right) \\
 $
Now the value of ${\left[ {10\left( {1 + .001} \right)} \right]^{1/3}}$
$
   = 10\left( {1 + 0.0003333 - 0.000001 + .......} \right) \\
   = 10(1 + 0.0003332) \\
   = 10 + 0.003332 \\
   = 10.0033 \\
 $
which is the value of ${\left( {1001} \right)^{1/3}}$

Hence, the cube root of 1001 is 10.0033.

Note: Since the calculations of the Binomial expansion require a lot of calculations so they have to be done very carefully.