Question

Find the cube root of 2.197. Find the cube root of -13824.

Answer 1

Hint: When the number is multiplied by itself three times, the result we got is the cube of the number. Alternatively, when the number is raised to the power of $\dfrac{1}{3}$, the result obtained is the cube root of the number. Cube root of $x$, is denoted as $\sqrt[3]{x}$ such that $\sqrt[3]{x} \times \sqrt[3]{x} \times \sqrt[3]{x} = x$.
To find the cube root of a number by factorization, first, find the prime factors of the number and make a group of triplets of the same numbers from the prime factors and then find their products.
For example, the prime factor of \[\left( c \right) = a \times a \times b \times b \times a \times b = \underline {\left[ {a \times a \times a} \right]} \times \underline {\left[ {b \times b \times b} \right]} = a \times b\]
In this question, we need to first evaluate the prime factors of the numbers and then group them in the number of three.

Complete step by step answer:
 Covert the decimal numbers into fractions for the ease to find the cube roots of the decimal numbers. So, 2.197 can be converted into a fraction number $\dfrac{{2197}}{{1000}}$.

Now, let us first find the cube root of 2197 by following the factorization method as:

\[
   13\underline {\left| {2197} \right.} \\
   13\underline {\left| {169} \right.} \\
   13\underline {\left| {13} \right.} \\
   1 \\
 \]

Hence, the number $2197 = 13 \times 13 \times 13 - - - - (i)$
Also, $1000 = 10 \times 10 \times 10 - - - - (ii)$

By equation (i) and (ii), we can say that the cube root of 2197 is 13, and that of 1000 is 10.
The cube root of 2.197 can be written as:
$\sqrt[3]{{2.197}} = \sqrt[3]{{\dfrac{{2197}}{{1000}}}} = \dfrac{{13}}{{10}} = 1.3$
Hence, the cube root of 2.197 is 1.3

Similarly, for the number -13824, first find the prime factors of the number 13824 as:

\[
   3\underline {\left| {13824} \right.} \\
   3\underline {\left| {4608} \right.} \\
   3\underline {\left| {1536} \right.} \\
   2\underline {\left| {512} \right.} \\
   2\underline {\left| {256} \right.} \\
   2\underline {\left| {128} \right.} \\
   2\underline {\left| {64} \right.} \\
   2\underline {\left| {32} \right.} \\
   2\underline {\left| {16} \right.} \\
   2\underline {\left| 8 \right.} \\
   2\underline {\left| 4 \right.} \\
   2\underline {\left| 2 \right.} \\
   \underline {\left| 1 \right.} \\
 \]
Hence, the number 13824 can be written as: $13824 = \left[ {3 \times 3 \times 3} \right] \times \left[ {2 \times 2 \times 2} \right] \times \left[ {2 \times 2 \times 2} \right] \times \left[ {2 \times 2 \times 2} \right]$
Moreover, the cube root of a negative number is always negative. So,
$\sqrt[3]{{ - 13824}} = - \sqrt[3]{{13824}} = - \left( {3 \times 2 \times 2 \times 2} \right) = - 24$

Hence, the cube root of -13824 is -24.

Note: The cube root of a number can either be found by using the estimation method or by the factorization method. But the best and easy method of finding cube roots is the factorization method as this has fewer calculations and saves time as well.