Question

Find the cube root of the following numbers by prime factorization method.
1).64
2).512
3).10648
4).27000
5).15625
6).13824
7).110592
8).46656
9).175616
10).91125

Answer 1

Hint: Do prime factorization of given numbers and make a group of 3 same numbers. Get the product of numbers from the group taken at once from each group. Apply only prime numbers for factorization of the given numbers, do not use any non-prime numbers for writing the factors of it.

Complete step-by-step answer:
The process of finding the cube root of any number is that we need to represent the given number in the form of products of the prime numbers and hence make a group of three same numbers and find the product of the numbers from each group and number taken at once only. So, let us proceed with these rules and get the cube root of the given rules.

64
So, let us do prime factorization by starting to divide numbers by 2. So, we get
\[\begin{align}
  & 2\left| 64 \right. \\
 & 2\left| 32 \right. \\
 & 2\left| 16 \right. \\
 & 2\left| 8 \right. \\
 & 2\left| 4 \right. \\
 & 2\left| 2 \right. \\
 & \,\,\,\left| 1 \right. \\
\end{align}\]
Hence, 64 can be represented in from of product of the factors as
\[64\ =\ \overline{2\times 2\times 2}\times \overline{2\times 2\times 2}\]
Now, we can observe the groups that are made with the same numbers of three times. So, taking the numbers from each group at once will the cube root of the given number.
Cube root of \[64\ =\ 2\times 2\ =\ 4\]

512
Prime factorization can be given as
\[\begin{align}
  & 2\left| 512 \right. \\
 & 2\left| 256 \right. \\
 & 2\left| 128 \right. \\
 & 2\left| 64 \right. \\
 & 2\left| 32 \right. \\
 & 2\left| 16 \right. \\
 & 2\left| 8 \right. \\
 & 2\left| 4 \right. \\
 & 2\left| 2 \right. \\
 & \,\,\,\left| 1 \right. \\
\end{align}\]
Hence, we get the representation in the form of factors as
\[512\ =\ \overline{2\times 2\times 2}\times \overline{2\times 2\times 2}\times \overline{2\times 2\times 2}\]
Cube root of 512 is given as
Cube root \[=\ 2\times 2\times 2\ =\ 8\]

10648
So, let us do prime factorization by starting to divide the number by 2. So, we get
\[\begin{align}
  & 2\left| 10648 \right. \\
 & 2\left| 5324 \right. \\
 & 2\left| 2662 \right. \\
 & 11\left| 1331 \right. \\
 & 11\left| 121 \right. \\
 & 11\left| 11 \right. \\
 & \,\,\ \ \left| 1 \right. \\
\end{align}\]
Now, we can observe the group that can have the same numbers of three times. So, taking the numbers from each group at once will the cube root of the given number.
\[10648\ =\overline{2\times 2\times 2}\times \overline{11\times 11\times 11}\]
Cube root of \[10648\ =\ 2\times 11\ =\ 22\]

27000

\[\begin{align}
  & 3\left| 27000 \right. \\
 & 3\left| 9000 \right. \\
 & 3\left| 3000 \right. \\
 & 2\left| 1000 \right. \\
 & 2\left| 500 \right. \\
 & 2\left| 250 \right. \\
 & 5\left| 125 \right. \\
 & 5\left| 25 \right. \\
 & 5\left| 5 \right. \\
 & \,\,\,\left| 1 \right. \\
\end{align}\]
\[27000\ =\ \overline{3\times 3\times 3}\times \overline{2\times 2\times 2}\times \overline{5\times 5\times 5}\]
Cube root of \[27000\ =\ 3\times 2\times 5\ =\ 30\]

13824
\[\begin{align}
  & 2\left| 13824 \right. \\
 & 2\left| 6912 \right. \\
 & 2\left| 3456 \right. \\
 & 2\left| 1728 \right. \\
 & 2\left| 864 \right. \\
 & 2\left| 432 \right. \\
 & 2\left| 216 \right. \\
 & 2\left| 108 \right. \\
 & 2\left| 54 \right. \\
 & 3\left| 27 \right. \\
 & 3\left| 9 \right. \\
 & 3\left| 3 \right. \\
 & \,\,\,\left| 1 \right. \\
\end{align}\]
\[13824\ =\ \overline{2\times 2\times 2}\times \overline{2\times 2\times 2}\times \overline{2\times 2\times 2}\times \overline{3\times 3\times 3}\]
Cube root \[=\ 2\times 2\times 2\times 3\ =\ 24\]

15625

\[\begin{align}
  & 5\left| 15625 \right. \\
 & 5\left| 3125 \right. \\
 & 5\left| 625 \right. \\
 & 5\left| 125 \right. \\
 & 5\left| 25 \right. \\
 & 5\left| 5 \right. \\
 & \,\,\,\left| 1 \right. \\
\end{align}\]
\[15625\ =\ \overline{5\times 5\times 5}\times \overline{5\times 5\times 5}\]
Cube root of \[15625\ =\ 5\times 5\ =\ 25\]

110592
\[\begin{align}
  & 2\left| 110592 \right. \\
 & 2\left| 55296 \right. \\
 & 2\left| 27648 \right. \\
 & 2\left| 13824 \right. \\
 & 2\left| 6912 \right. \\
 & 2\left| 3456 \right. \\
 & 2\left| 1728 \right. \\
 & 2\left| 864 \right. \\
 & 2\left| 432 \right. \\
 & 2\left| 216 \right. \\
 & 2\left| 108 \right. \\
 & 2\left| 54 \right. \\
 & 3\left| 27 \right. \\
 & 3\left| 9 \right. \\
 & 3\left| 3 \right. \\
 & \,\,\,\left| 1 \right. \\
\end{align}\]
\[110592\ =\ \overline{2\times 2\times 2}\times \overline{2\times 2\times 2}\times \overline{2\times 2\times 2}\times \overline{2\times 2\times 2}\times \overline{3\times 3\times 3}\]
Cube root \[=\ 2\times 2\times 2\times 2\times 3\ =\ 48\]

46656
\[\begin{align}
  & 2\left| 46656 \right. \\
 & 2\left| 23328 \right. \\
 & 2\left| 11664 \right. \\
 & 2\left| 5832 \right. \\
 & 2\left| 2916 \right. \\
 & 2\left| 1458 \right. \\
 & 3\left| 729 \right. \\
 & 3\left| 243 \right. \\
 & 3\left| 81 \right. \\
 & 3\left| 27 \right. \\
 & 3\left| 9 \right. \\
 & 3\left| 3 \right. \\
 & \,\,\,\left| 1 \right. \\
\end{align}\]
\[46656\ =\ \overline{2\times 2\times 2}\times \overline{2\times 2\times 2}\times \overline{3\times 3\times 3}\times \overline{3\times 3\times 3}\]
Cube root \[=\ 2\times 2\times 3\times 3\ =\ 36\]

 175616
\[\begin{align}
  & 2\left| 175616 \right. \\
 & 2\left| 87808 \right. \\
 & 2\left| 43904 \right. \\
 & 2\left| 21952 \right. \\
 & 2\left| 10976 \right. \\
 & 2\left| 5488 \right. \\
 & 2\left| 2744 \right. \\
 & 2\left| 1372 \right. \\
 & 2\left| 686 \right. \\
 & 7\left| 343 \right. \\
 & 7\left| 49 \right. \\
 & 7\left| 7 \right. \\
 & \,\,\,\left| 1 \right. \\
\end{align}\]
\[175616\ =\ \overline{2\times 2\times 2}\times \overline{2\times 2\times 2}\times \overline{2\times 2\times 2}\times \overline{7\times 7\times 7}\]
Cube root \[=\ 2\times 2\times 2\times 7\ =\ 56\]

91125
\[\begin{align}
  & 5\left| 91125 \right. \\
 & 5\left| 18225 \right. \\
 & 5\left| 3645 \right. \\
 & 3\left| 729 \right. \\
 & 3\left| 243 \right. \\
 & 3\left| 81 \right. \\
 & 3\left| 27 \right. \\
 & 3\left| 9 \right. \\
 & 3\left| 3 \right. \\
 & \,\,\,\left| 1 \right. \\
\end{align}\]
\[91125\ =\ \overline{5\times 5\times 5}\times \overline{3\times 3\times 3}\times \overline{3\times 3\times 3}\]
Cube root of \[91125\ =\ 5\times 3\times 3\ =\ 45\]

Note: Take the factors of the given numbers as only prime numbers. Don’t include any other number, which is not a prime. And always make a group of factors by taking 3 same numbers in each group. And take the number only once from each and multiply them to get a cube root.