Question

Find the cube root of the following 10648, 42.875.

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 the power of $\dfrac{1}{3}$, the result obtained is
the cube root of number. Cube root of $x$, is \[\sqrt[3]{x} = \sqrt x \times \sqrt x \times \sqrt x = x\]
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, prime factor of \[\left( c \right) = a \times a \times b \times b \times a \times b = \underline
{\left[ {a \times a \times a} \right]} \times \underline {\left[ {b \times b \times b} \right]} = a \times b\]
Complete step by step solution:
Let’s find the cube root of the number using factorization method first we will factorize the given
numbers.
\[
2\underline {\left| {10648} \right.} \\
2\underline {\left| {5324} \right.} \\
2\underline {\left| {2662} \right.} \\
11\underline {\left| {1331} \right.} \\
11\underline {\left| {121} \right.} \\
11\underline {\left| {11} \right.} \\
1 \\
\]
Hence we can write \[\left( {10648} \right) = 2 \times 2 \times 2 \times 11 \times 11 \times 11 \times 1\]
Now make triplet group of the factors of 10648:
\[
\left( {10648} \right) = \underline {\left[ {2 \times 2 \times 2} \right]} \times \left[ {\underline {11
\times 11 \times 11} } \right] \\
= 2 \times 11 = 22 \\
\]
Hence the cube root of 10648 is 22.

Similarly, follow the same process to find the cube root of\[42.875\]; here, we can see the number has a
decimal place in it, so we convert that decimal into a fraction. To convert the decimal number into a
fraction, we write the number into \[\dfrac{a}{b}\]form; hence we can write\[42.875 = \dfrac{a}{b} =
\dfrac{{42875}}{{1000}}\], now we will find cube root separately, we know \[1000 = {\left( {10}
\right)^3}\]hence its cube root will be \[\sqrt[3]{{{{\left( {10} \right)}^3}}} = {\left( {10} \right)^{3
\times \dfrac{1}{3}}} = 10\].
Let’s factorize
\[
7\underline {\left| {42875} \right.} \\
7\underline {\left| {6125} \right.} \\
7\underline {\left| {875} \right.} \\
5\underline {\left| {125} \right.} \\
5\underline {\left| {25} \right.} \\
5\underline {\left| 5 \right.} \\
1 \\
\]
So we write \[\left( {42875} \right) = 7 \times 7 \times 7 \times 5 \times 5 \times 5 \times 1\]
Now make triplets group of the factors of 42875:
\[
\left( {42875} \right) = 7 \times 7 \times 7 \times 5 \times 5 \times 5 \times 1 = \left[ {\underline {7
\times 7 \times 7} } \right] \times \left[ {\underline {5 \times 5 \times 5} } \right] \\
= 7 \times 5 = 35 \\
\]
Hence cube root of 42.875:
\[
\left( {42.875} \right) = \dfrac{{42875}}{{1000}} \\
\sqrt[3]{{\left( {42.875} \right)}} = \dfrac{{\sqrt[3]{{42875}}}}{{\sqrt[3]{{1000}}}} \\
= \dfrac{{35}}{{10}} = 3.5 \\

\]
Note: In the case of decimal numbers, always try to convert the number into a fraction; it makes it easy
to find a cube root of the number. The cube root of a number can either be found by using the
estimation method or by factorization method.