Question

without actually calculating the cubes find the value of each of the following (-28)^3+(12)^3+(16)^3 ​

Answer 1

-16128

Explanation:

This expression can be calculated by algebraic means, whose process is described below:

1)
`(-28)^(3)+(12)^(3)+(16)^(3)` Given.

2)
`(-12-16)^(3) + (12)^(3)+(16)^(3)` Definition of addition.

3)
`(-12)^(3) + 3\cdot (-12)^(2)\cdot (-16)+3\cdot (-12)\cdot (-16)^(2)+(-16)^(3)+(12)^(3)+(16)^(3)` Cubic perfect binomial.

4)
`(12)^(3)+[(-1)\cdot (12)]^(3)+(16)^(3) + [(-1)\cdot (16)]^(3)+3 \cdot (-12)^(2)\cdot (-16) + 3\cdot (-12)\cdot (-16)^(2)` Commutative property/
`(-x)\cdot y = -x\cdot y`

5)
`(12)^(3) + (-1)^(3)\cdot (12)^(3) + 16^(3) +(-1)^(3)\cdot (16)^(3) + (-3)\cdot [(-12)^(2)\cdot (16) +(-16)^(2)\cdot (12)]` Distributive property/
`(-x)\cdot y = -x\cdot y`/
`x^(n)\cdot y^(n) = (x\cdot y)^(n)`

6)
`(12)^(3) + [-(12)^(3)]+(16)^(3) + [-(16)^(3)]+ (-3)\cdot [(-12)^(2)\cdot (16)+(-16)^(2)\cdot (12)]`
`(-x)\cdot y = -x\cdot y`

7)
`(-3)\cdot [(-12)^(2)\cdot (16) + (-16)^(2)\cdot (12)]` Existence of the additive inverse/Modulative property for addition.

8)
`(-3) \cdot [(12)^(2)\cdot (16)+(16^(2))\cdot (12)]`
`x^(n)\cdot y^(n) = (x\cdot y)^(n)`/
`(-x)\cdot (-y) = x\cdot y`

9)
`(-3)\cdot (12)\cdot (16)\cdot (12+16)` Distributive property.

10)
`-16128`
`(-x)\cdot y = -x\cdot y`/Definition of sum/Definition of multiplication/Result