Cube Root Calculation for 1728

Understanding the concept of a cube root is fundamental in mathematics. The cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 8 is 2, because $2 \times 2 \times 2 = 8$. We denote the cube root using the symbol $\sqrt[3]{}$.

Finding the Cube Root of 1728

To find the cube root of 1728, we can use the prime factorization method. This method involves breaking down the number into its prime factors and then grouping them in sets of three.

Step-by-Step Prime Factorization

1. Start with the number: We need to find the prime factors of 1728.
2. Divide by the smallest prime number: Begin by dividing 1728 by the smallest prime number, which is 2, as 1728 is an even number.

From the factorization, we can write 1728 as a product of its prime factors:

$\qquad 1728 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3$

Grouping Prime Factors for Cube Root

For the cube root, we group identical prime factors in sets of three:

• Group 1: $(2 \times 2 \times 2)$
• Group 2: $(2 \times 2 \times 2)$
• Group 3: $(3 \times 3 \times 3)$

So, we can rewrite the prime factorization as:

$\qquad 1728 = (2^3) \times (2^3) \times (3^3)$

Now, to find the cube root of 1728, we take one number from each group:

$\qquad \sqrt[3]{1728} = \sqrt[3]{2^3 \times 2^3 \times 3^3}$

$\qquad \sqrt[3]{1728} = 2 \times 2 \times 3$

$\qquad \sqrt[3]{1728} = 4 \times 3$

$\qquad \sqrt[3]{1728} = 12$

Verification of the Cube Root

To verify our answer, we can cube the result, 12:

$\qquad 12^3 = 12 \times 12 \times 12$

$\qquad 12^3 = 144 \times 12$

$\qquad 12^3 = 1728$

Since $12^3$ equals 1728, our calculation is correct. Therefore, the cube root of 1728 is 12.