Cube Root through Prime Factorisation Method
In this section, we learn how to find the cube root of a number using the prime factorisation method. The cube root is determined by expressing a number as the product of its prime factors and identifying how many times each factor is used in groups of three.
Two significant examples illustrated include:
1. Finding the cube root of 3375: The prime factorisation is done to express 3375 as \( 3375 = 3^3 \times 5^3 = (3 \times 5)^3 \). Hence, the cube root of 3375 is \( 3 \sqrt[3]{3375} = 15 \).
2. Finding the cube root of 74088: Following similar steps, we find \( 74088 = 2^3 \times 3^3 \times 7^3 = (2 \times 3 \times 7)^3 \), which leads to a cube root of \( \sqrt[3]{74088} = 42 \).
This method is significant as it not only offers a systematic approach to finding cube roots but also reinforces understanding of prime factorisation in mathematical terms.
Example 7: Find the cube root of 54000.
Solution: Prime factorisation of 54000 is
$$54000 = 2^3 \times 3^3 \times 5^3$$.
So,
$$\sqrt[3]{54000} = 2 \times 3 \times 5 = 30.$$
Similar Question:
Example 8: Find the cube root of 729000.
Solution: Prime factorisation of 729000 is
$$729000 = 3^6 \times 2^3 \times 5^3.$$
So,
$$\sqrt[3]{729000} = 3^2 \times 2 \times 5 = 45.$$