Calculation of (49^3 - 22^3 - 27^3)

To calculate the expression (49^3 - 22^3 - 27^3) without actually performing the calculation, we can use a concept called the difference of cubes.

Difference of Cubes Formula:
The difference of cubes formula states that for any two numbers 'a' and 'b', the expression (a^3 - b^3) can be factored as (a - b)(a^2 + ab + b^2).

Step 1: Factorizing 49^3:
We can factorize 49^3 using the difference of cubes formula:
49^3 = (49 - 22)(49^2 + 49*22 + 22^2)

Step 2: Simplifying (49 - 22):
49 - 22 = 27

Step 3: Simplifying 49^2 + 49*22 + 22^2:
To simplify this expression, we can calculate the squares and products separately:
- 49^2 = 2401
- 49*22 = 1078
- 22^2 = 484

Now, we can add these values together:
49^2 + 49*22 + 22^2 = 2401 + 1078 + 484 = 3963

Step 4: Combining the simplified terms:
Now that we have simplified both terms, we can substitute them back into the original expression:
49^3 - 22^3 - 27^3 = (49 - 22)(49^2 + 49*22 + 22^2) - 27^3
= 27 * 3963 - 27^3

Step 5: Simplifying the expression:
To simplify further, we can factor out the common factor of 27:
27 * 3963 - 27^3 = 27(3963 - 27^2)

Step 6: Simplifying 3963 - 27^2:
To simplify this expression, we can calculate the square:
27^2 = 729

Now we can subtract the square from 3963:
3963 - 729 = 3234

Step 7: Final Result:
Substituting the simplified term back into the expression:
27(3963 - 27^2) = 27 * 3234 = 87258

Therefore, without actually performing the calculation, the value of (49^3 - 22^3 - 27^3) is 87258.

u have to use the frmula (-a ) + (-b³) + ( c³) = ( a+ b+ c)(a²+b²+c²-ab-bc-ca) + 3abc