Question

Making use of the cube root table, find the cube root of \[8.65\] correct to three decimal places.

Answer 1

Hint: We will find the nearest whole number of the given number so that it will be easy for us to find the cube roots by using the table. We will find the fractional form of the given decimal value for making our approximation correct to three decimal points.

Complete step-by-step answer:
Here we are asked to find the cube root of the number \[8.65\]
So we have \[\sqrt[3]{{8.65}}\]
Now we will write this number \[8.65\] in fractional form, which will be equal to \[\dfrac{{865}}{{100}}\]
\[\sqrt[3]{{8.65}} = \sqrt[3]{{\dfrac{{865}}{{100}}}}\]
Let’s find the cube root of $ 100 $
 $ \sqrt[3]{{100}} = 4.642 $
Now, we have \[860 < 865 < 870\]
So, let’s find the cube roots of three of them individually
 $\Rightarrow \sqrt[3]{{860}} = 9.510 $ --(1)
Then let’s find the roots of \[870\]
 $\Rightarrow \sqrt[3]{{870}} = 9.546 $ --(2)
Now we will try to find the difference in values of (1) and (2)
Difference in values \[ = 870 - 860 = 10\]
Now we will find the difference in cube roots values
Difference in cube roots value \[ = 9.546 - 9.510 = 0.036\]
Similarly we will do it for \[865\] and \[860\] respectively
Difference between values \[ = 865 - 860 = 5\]
Now the difference in cube roots can be evaluated.
Difference in cube roots values \[ = 100.036 \times 5 = 0.018\]
\[\Rightarrow \sqrt[3]{{865}} = 9.510 + 0.018 = 9.528\]
Now we have,
\[\Rightarrow \sqrt[3]{{8.65}} = \sqrt[3]{{\dfrac{{865}}{{100}}}} = \dfrac{{9.528}}{{4.642}}\]
\[ \Rightarrow \sqrt[3]{{8.65}} = \dfrac{{9528}}{{4642}} = 2.052\]
Hence, we got the answer as \[2.052\] correct to three decimal points.
So, the correct answer is “ \[2.052\]”.

Note: Always convert the decimal numbers to fractional numbers as it reduces the calculation to a smaller extent and also helps to find an accurate number. Making calculations requires a lot of effort and rough work, as calculation is the basis of such numbers, so do it carefully taking care of each and every digit.