Question

1. ((v^6 + w^6)^2 = (v^6 - w^6)^2 + (2v^3w^3)^2)

Explanation:

Step1: Expand left side

Using the formula \((a + b)^2 = a^2 + 2ab + b^2\), where \(a = v^6\) and \(b = w^6\), we get:
\((v^6 + w^6)^2 = (v^6)^2 + 2 \cdot v^6 \cdot w^6 + (w^6)^2 = v^{12} + 2v^6w^6 + w^{12}\)

Step2: Expand right side

First, expand \((v^6 - w^6)^2\) using \((a - b)^2 = a^2 - 2ab + b^2\) (where \(a = v^6\), \(b = w^6\)):
\((v^6 - w^6)^2 = (v^6)^2 - 2 \cdot v^6 \cdot w^6 + (w^6)^2 = v^{12} - 2v^6w^6 + w^{12}\)

Then, expand \((2v^3w^3)^2\) using \((ab)^2 = a^2b^2\) (where \(a = 2\), \(b = v^3w^3\)):
\((2v^3w^3)^2 = 2^2 \cdot (v^3)^2 \cdot (w^3)^2 = 4v^6w^6\)

Now, add the two expanded terms of the right side:
\((v^6 - w^6)^2 + (2v^3w^3)^2 = (v^{12} - 2v^6w^6 + w^{12}) + 4v^6w^6\)
Simplify the like terms:
\(= v^{12} + (-2v^6w^6 + 4v^6w^6) + w^{12} = v^{12} + 2v^6w^6 + w^{12}\)

Step3: Compare both sides

Left side: \(v^{12} + 2v^6w^6 + w^{12}\)
Right side: \(v^{12} + 2v^6w^6 + w^{12}\)
Since both sides are equal, the equation holds true.

Answer:

The equation \((v^6 + w^6)^2 = (v^6 - w^6)^2 + (2v^3w^3)^2\) is an identity (it holds for all values of \(v\) and \(w\)).