Calculating the Difference of Cubes: $27^3 - 22^3$

To find the value of $27^3 - 22^3$, we can use the algebraic identity for the difference of cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

1. Identify $a$ and $b$:

   • $a = 27$
   • $b = 22$

2. Calculate the difference $(a - b)$:

   $27 - 22 = 5$

3. Calculate the terms $a^2$, $ab$, and $b^2$:

   • $a^2 = 27^2 = 729$
   • $ab = 27 \times 22 = 594$
   • $b^2 = 22^2 = 484$

4. Calculate the sum $(a^2 + ab + b^2)$:

   $729 + 594 + 484 = 1807$

5. Multiply the results from step 2 and step 4:

   $5 \times 1807 = 9035$

Alternatively, calculate the cubes directly:

• $27^3 = 19683$
• $22^3 = 10648$
• $27^3 - 22^3 = 19683 - 10648 = 9035$

Therefore, the value of $27^3 - 22^3$ is $9035$.