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

Assign variables: a = 28 , b = − 15 , and 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 Now, let's check if a + b + c = 0 . We have 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 .

Applying the Identity Using the identity, we have a 3 + b 3 + c 3 = 3 ab c = 3 ( 28 ) ( − 15 ) ( − 13 ) .

Calculating the Result Now, we calculate the value of 3 ( 28 ) ( − 15 ) ( − 13 ) . The result of the operation is 16380 .

Final Answer Therefore, ( 28 ) 3 + ( − 15 ) 3 + ( − 13 ) 3 = 16380 .


Examples
Consider a scenario where you need to calculate the total volume change of three materials when they are heated or cooled. If the volume changes can be represented as cubes of numbers and their sum is zero, you can use the identity a 3 + b 3 + c 3 = 3 ab c to simplify the calculation. This can be applied in engineering or physics to quickly estimate the combined effect without computing individual cubes, saving time and resources. For instance, if material A expands by 28 units cubed, material B contracts by 15 units cubed, and material C contracts by 13 units cubed, the net volume change can be easily calculated using this identity.

Answered by GinnyAnswer | 2025-07-08

The value of ( 28 ) 3 + ( − 15 ) 3 + ( − 13 ) 3 can be determined using the identity a 3 + b 3 + c 3 = 3 ab c , where the sum of the numbers is zero. By calculating 3 ( 28 ) ( − 15 ) ( − 13 ) , we find the result is 16380 . Thus, the answer is 16380 .
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Answered by Anonymous | 2025-08-03