Finding Cube Roots

To find the cube root of a number, we can use prime factorisation and group the identical factors. Let us examine the prime factorisation of $1728$ to find its cube root. First, we break down $1728$ into its prime factors, resulting in $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3$. Next, we must organise these prime factors into identical groups to determine if it is a perfect cube. Writing this with exponents, we get $2^3 \times 2^3 \times 3^3$, which shows each prime factor appears the correct number of times. By taking one number from each triplet, the cube root calculation becomes $2 \times 2 \times 3$. Multiplying these factors together, we conclude that the cube root of $1728$ is .