(1-x^2-y^2)^2/2*(x^2+y^2)
You asked:
Evaluate the expression: \(\frac{{\left( 1 - {x}^{2} - {y}^{2} \right)}^{2}}{2} \cdot \left({x}^{2} + {y}^{2}\right)\)
MathBot Answer:
Evaluated
\(\displaystyle \frac{{\left( 1 - {x}^{2} - {y}^{2} \right)}^{2}}{2} \cdot \left({x}^{2} + {y}^{2}\right) = \frac{\left(x^{2} + y^{2}\right) \left(- x^{2} - y^{2} + 1\right)^{2}}{2} \)
Expanded
\[\frac{{\left( 1 - {x}^{2} - {y}^{2} \right)}^{2}}{2} \cdot \left({x}^{2} + {y}^{2}\right) = \frac{x^{6}}{2} + \frac{3 x^{4} y^{2}}{2} - x^{4} + \frac{3 x^{2} y^{4}}{2} - 2 x^{2} y^{2} + \frac{x^{2}}{2} + \frac{y^{6}}{2} - y^{4} + \frac{y^{2}}{2}\]
Factored
\[\frac{{\left( 1 - {x}^{2} - {y}^{2} \right)}^{2}}{2} \cdot \left({x}^{2} + {y}^{2}\right) = \frac{\left(x^{2} + y^{2}\right) \left(x^{2} + y^{2} - 1\right)^{2}}{2}\]