Question

Find the cube root of the given number by prime factorization method: $13824$

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$\sqrt[3]{x} \times \sqrt[3]{x} \times \sqrt[3]{x} = x$.

Complete step-by-step answer:
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 \[\left( c \right) = a \times a \times b \times b \times a \times b = \underline {\left[ {a \times a \times a} \right]} \times \underline {\left[ {b \times b \times b} \right]} = a \times b\]
Prime factorization is the process of finding which prime numbers can be multiplied together to make the original number, where prime numbers are the numbers greater than and which are not the product of any two smaller natural numbers. Prime numbers are the numbers which are either divisible by 1 or by itself. (e.g. 2, 3, 5, 7, 11)
Let’s find the cube root of the number using factorization method first we will factorize the given numbers only by the prime numbers.
\[
   2\underline {\left| {13824} \right.} \\
   2\underline {\left| {6912} \right.} \\
   2\underline {\left| {3456} \right.} \\
   2\underline {\left| {1728} \right.} \\
   2\underline {\left| {864} \right.} \\
   2\underline {\left| {432} \right.} \\
   2\underline {\left| {216} \right.} \\
   2\underline {\left| {108} \right.} \\
   2\underline {\left| {54} \right.} \\
   3\underline {\left| {27} \right.} \\
   3\underline {\left| 9 \right.} \\
   3 \\
 \]
Hence we can write \[\left( {13824} \right) = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3\]
Now make triplet group of the factors of 13824:
\[\left( {13824} \right) = \underline {2 \times 2 \times 2} \times \underline {2 \times 2 \times 2} \times \underline {2 \times 2 \times 2} \times \underline {3 \times 3 \times 3} \]
\[
  \sqrt[3]{{\left( {13824} \right)}} = 2 \times 2 \times 2 \times 3 \\
   = 24 \\
 \]
Hence the cube root of 13824 is 24.
To check weather 24 is the cube root of 13824 or not we will find the cube of the number,
\[{\left( {24} \right)^3} = 24 \times 24 \times 24 = 13824\]
Hence we can say 24 is the cube root of 13824

Note: The cube root of a number can either be found by using the estimation method or by 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.