Without actually calculating cubes, find the value of $(28)^3+(-15)^3+(-13)^3$

Assign variables: a = 28 , b = − 15 , c = − 13 .
Verify that a + b + c = 0 .
Apply the identity a 3 + b 3 + c 3 = 3 ab c .
Calculate the result: 3 ( 28 ) ( − 15 ) ( − 13 ) = 16380 ​ $.

Explanation

Understanding the Problem We are given the expression ( 28 ) 3 + ( − 15 ) 3 + ( − 13 ) 3 . We want to find its value without directly calculating the cubes.

Assigning Variables Let a = 28 , b = − 15 , and c = − 13 . Then the expression becomes a 3 + b 3 + c 3 .

Checking the Condition We check if a + b + c = 0 : a + b + c = 28 + ( − 15 ) + ( − 13 ) = 28 − 15 − 13 = 0 Since a + b + c = 0 , we can use the identity a 3 + b 3 + c 3 = 3 ab c .

Calculating the Result Now we calculate 3 ab c : 3 ab c = 3 ( 28 ) ( − 15 ) ( − 13 ) = 16380 Therefore, ( 28 ) 3 + ( − 15 ) 3 + ( − 13 ) 3 = 16380 .

Final Answer Thus, the value of the expression ( 28 ) 3 + ( − 15 ) 3 + ( − 13 ) 3 is 16380 .


Examples
This mathematical concept can be applied in engineering to calculate the net effect of forces or stresses acting in different directions. For example, if three forces are acting on a point and their sum is zero, the sum of the cubes of their magnitudes can be easily calculated using the identity a 3 + b 3 + c 3 = 3 ab c , simplifying complex calculations and aiding in structural analysis.

Answered by GinnyAnswer | 2025-07-08

The value of ( 28 ) 3 + ( − 15 ) 3 + ( − 13 ) 3 can be calculated using the identity a 3 + b 3 + c 3 = 3 ab c since their sum is zero. Assigning a = 28 , b = − 15 , and c = − 13 leads to the result of 16380 . Therefore, the final answer is 16380 .
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Answered by Anonymous | 2025-07-12