Question

Without actually calculating the cube, find the value of the following:

`(-8/15)^3 + (1/3)^3 + (1/5)^3`

Answer 1

Given: `(-8/15)^3 + (1/3)^3 + (1/5)^3`

Step-wise calculation:

1. Notice the sum of the three numbers inside the cubes:

`-8/15 + 1/3 + 1/5 = -8/15 + 5/15 + 3/15`

`-8/15 + 1/3 + 1/5 = 0`

2. Since a + b + c = 0, the identity for cubes states:

a³ + b³ + c³ = 3abc

3. Here, `a = -8/15, b = 1/3, c = 1/5`

4. Calculate the product (abc):

`abc = (-8/15) xx 1/3 xx 1/5`

`abc = - 8/225`

5. Using the identity:

`a^3 + b^3 + c^3 = 3 xx (-8/225)`

`a^3 + b^3 + c^3 = - 24/225`

`a^3 + b^3 + c^3 = - 8/75`