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**Hint:**Consider the given number 833 between 830 and 840. Take cube root of all the three numbers and write it in the form ${{\left( 830 \right)}^{\dfrac{1}{3}}} < {{\left( 833 \right)}^{\dfrac{1}{3}}} < {{\left( 840 \right)}^{\dfrac{1}{3}}}$. Using the cube root table find the cube roots of 830 and 840. Observe the difference in the cube roots of 830 and 840 for the difference of 10 between them. Use the unitary method to find the difference in the cube roots of 830 and 833 for the difference of 10 between them. Finally, add the obtained difference with the cube root of 830 to get the answer.

**Complete step by step answer:**

Here we have been asked to find the cube root of 833 correct up to three decimal places using the cube root table. We can consider 833 between 830 and 840 so we have,

$\Rightarrow 830 < 833 < 840$

Taking cube root of all the three numbers we get,

$\Rightarrow {{\left( 830 \right)}^{\dfrac{1}{3}}} < {{\left( 833 \right)}^{\dfrac{1}{3}}} < {{\left( 840 \right)}^{\dfrac{1}{3}}}$

Now, using the cube root table we have the values ${{\left( 830 \right)}^{\dfrac{1}{3}}}=9.398$ and ${{\left( 840 \right)}^{\dfrac{1}{3}}}=9.435$, so we get,

$\Rightarrow 9.398 < {{\left( 833 \right)}^{\dfrac{1}{3}}} < 9.435$

We can see that when the difference between the numbers 840 and 830 is 10, the difference in their cube roots is 9.435 – 9.398 = 0.037. Therefore, when we will consider the numbers 833 and 830 difference between them is 3 so using the unitary method we have the difference between their cube roots as: -

$\begin{align}

& \Rightarrow {{\left( 833 \right)}^{\dfrac{1}{3}}}-{{\left( 830 \right)}^{\dfrac{1}{3}}}=\dfrac{0.037}{10}\times 3 \\

& \Rightarrow {{\left( 833 \right)}^{\dfrac{1}{3}}}-9.398=0.0111 \\

& \Rightarrow {{\left( 833 \right)}^{\dfrac{1}{3}}}=9.4091 \\

\end{align}$

Rounding off the number up to three decimal places we get,

$\therefore {{\left( 833 \right)}^{\dfrac{1}{3}}}=9.409$

Hence, the cube root of 833 is 9.409 correct up to three places of decimal.

**Note:**When you observe the cube root table you will observe certain columns. In the first column we have certain numbers represented as x, in the second column we have the value ${{\left( x \right)}^{\dfrac{1}{3}}}$, in the third column we have ${{\left( 10x \right)}^{\dfrac{1}{3}}}$ and in the fourth column we have ${{\left( 100x \right)}^{\dfrac{1}{3}}}$. From here you may think about the reason for considering the relation 830 < 833 < 840. We can find the values of cube roots of 10 times and 100 times a number from the cube root table directly and not a number which does not end with the digit 0 like 833.

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