Example: Find $\sqrt[3]{2744}$

1. Factorize: $2744$ breaks down into $2 \times 2 \times 2 \times 7 \times 7 \times 7$.

2. Group Triplets: Write it as $\sqrt[3]{(2 \times 2 \times 2) \times (7 \times 7 \times 7)}$.

3. Simplify: Pull one number from each triplet: $2 \times 7 = \mathbf{14}$.

Example: Find $\sqrt[3]{35937}$

1. Factorize: $35937$ breaks down into $3 \times 3 \times 3 \times 11 \times 11 \times 11$.

2. Group Triplets: $\sqrt[3]{(3 \times 3 \times 3) \times (11 \times 11 \times 11)}$.

3. Simplify: $3 \times 11 = \mathbf{33}$.

Essential Properties of Cube Roots

Understanding these three properties allows you to solve negative numbers, multiplications, and fractions with ease.

1. Cube Root of a Negative Number: $\sqrt[3]{-a} = -\sqrt[3]{a}$

The cube root of a negative number is always negative.

• Example: $\sqrt[3]{-8} = -\sqrt[3]{8} = -2$.

• Example: $\sqrt[3]{-1728} = -12$.

2. Product Property: $\sqrt[3]{a} \times \sqrt[3]{b} = \sqrt[3]{ab}$

You can combine two cube roots being multiplied into a single root.

• Example: $\sqrt[3]{2} \times \sqrt[3]{4} = \sqrt[3]{2 \times 4} = \sqrt[3]{8} = 2$.

• Advanced Example: To find $\sqrt[3]{140 \times 2450}$, multiply them to get $\sqrt[3]{343000}$. Since $\sqrt[3]{343} = 7$ and $\sqrt[3]{1000} = 10$, the answer is $7 \times 10 = \mathbf{70}$.

3. Quotient Property: $\frac{\sqrt[3]{a}}{\sqrt[3]{b}} = \sqrt[3]{\frac{a}{b}}$

Similarly, you can divide the numbers under a single cube root symbol.

• Example: $\frac{\sqrt[3]{108}}{\sqrt[3]{4}} = \sqrt[3]{\frac{108}{4}} = \sqrt[3]{27} = 3$.

• Fraction Example: $\sqrt[3]{-\frac{2197}{9261}} = -\frac{\sqrt[3]{2197}}{\sqrt[3]{9261}} = \mathbf{-\frac{13}{21}}$.