Question

Find the cube root of \[12167\] by prime factorization.

Answer 1

Hint: We can find the cube root of a given number without use of a calculator by doing factorization. Cube root of a number is denoted as \[\sqrt[3]{x}\]. Prime factorization is the method of factoring a number in terms of prime numbers, with prime numbers as the variables.

Complete step-by-step answer:
The prime factorisation approach can be used to find prime factors. If we want to find the prime factors of a given number, we must divide it by the least prime number, which leaves no remainder. Divide the resulting quotient by the least prime number and repeat the process until the quotient equals \[1\].
For example, \[12\]can be factorized in prime numbers as \[2 \times 2 \times 3 = {2^2} \times 3\]. Similarly, \[24\]can be factorized in prime numbers as \[2 \times 2 \times 2 \times 3 = {2^3} \times 3\].
A number's cube root is a value that, when multiplied by itself three times, returns the original value. For example, the cube root of \[27\]will be \[\sqrt[3]{{3 \times 3 \times 3}} = 3\].
To find the cube root of \[12167\]by prime factorization, we follow the following steps:
Step 1: Find out the prime number that when multiplied three times, provides us with a given number. In our case it will be \[23\] i.e.
\[23 \times 23 \times 23 = 12167\]
Step 2: Find the cube root of the numbers found in Step 1. We will get-
\[\sqrt[3]{{23 \times 23 \times 23}} = \sqrt[3]{{{{23}^3}}} = {23^{3(\dfrac{1}{3})}} = {23^1} = 23\]
Hence, the cube root of \[12167\] is \[23\] by prime factorization method.

Note: When prime numbers are multiplied by any natural or whole number (except 0), composite numbers are generated. Essentially, prime factorization is used to factorise and locate prime factors for composite numbers. This approach can also be used to calculate the HCF (Highest Common Factor) and LCM (Least Common Multiple) of a set of numbers.