Cube Roots

Definition

The cube root of a number $x$ is that number which, when multiplied by itself three times, gives $x$. Denoted as $\sqrt[3]{x}$.

Prime Factorization Method

To find the cube root, we factorize the number and group factors in triplets (groups of 3).

Example: Find $\sqrt[3]{3375}$

1. Prime Factorization: $3375 = 3 \times 3 \times 3 \times 5 \times 5 \times 5$

2. Group in Triplets: $3375 = (3 \times 3 \times 3) \times (5 \times 5 \times 5)$

3. Take one from each group: $\sqrt[3]{3375} = 3 \times 5$

4. Result: $15$

Estimation Method

For perfect cubes like 12167:

1. Group digits: Starting from the right, group in 3s. $\underline{12}, \underline{167}$.
2. First group (167): Ends in 7. Therefore, the cube root ends in 3 (since $3^3=27$).
3. Second group (12): Find the largest cube less than 12. $2^3 = 8$ and $3^3 = 27$. So, the ten’s digit is 2.
4. Result: 23.

Check: $23 \times 23 \times 23 = 12167$. Correct.