Question

let x and y represent natural numbers. prove that the following equation is true for all x and y values. show your work or explain your answer. ((x^2 + y^2)^2 - (x^2 - y^2)^2 = (2xy)^2)

Explanation:

Step1: Expand left-hand side terms

First, expand $(x^2+y^2)^2$ and $(x^2-y^2)^2$ using the square of a sum/difference formula $(a\pm b)^2=a^2\pm2ab+b^2$:
$$(x^2+y^2)^2 = x^4 + 2x^2y^2 + y^4$$
$$(x^2-y^2)^2 = x^4 - 2x^2y^2 + y^4$$

Step2: Subtract the expanded terms

Subtract the second expanded expression from the first:
$$(x^4 + 2x^2y^2 + y^4) - (x^4 - 2x^2y^2 + y^4)$$
$$=x^4 + 2x^2y^2 + y^4 - x^4 + 2x^2y^2 - y^4$$

Step3: Simplify the left-hand side

Combine like terms to simplify:
$$(x^4-x^4)+(2x^2y^2+2x^2y^2)+(y^4-y^4)=4x^2y^2$$

Step4: Expand the right-hand side

Expand $(2xy)^2$ using the power of a product rule $(ab)^n=a^nb^n$:
$$(2xy)^2=2^2x^2y^2=4x^2y^2$$

Step5: Compare both sides

Both the simplified left-hand side and expanded right-hand side are equal.

Answer:

The equation $(x^2 + y^2)^2 - (x^2 - y^2)^2 = (2xy)^2$ is true for all natural numbers $x$ and $y$, as both sides simplify to $4x^2y^2$.