Solve f(x)=(x/sqrt{x^{2+y^{2}}})^{2}+y^2/(sqrt{x^2+y^2)^2}

$$f ( x ) = ( \frac { x } { \sqrt { x ^ { 2 } + y ^ { 2 } } } ) ^ { 2 } + \frac { y ^ { 2 } } { ( \sqrt { x ^ { 2 } + y ^ { 2 } } ) ^ { 2 } }$$

Evaluate

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Solution Steps

To raise \frac{x}{\sqrt{x^{2}+y^{2}}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{x^{2}}{\left(\sqrt{x^{2}+y^{2}}\right)^{2}}+\frac{y^{2}}{\left(\sqrt{x^{2}+y^{2}}\right)^{2}}$$

Calculate \sqrt{x^{2}+y^{2}} to the power of 2 and get x^{2}+y^{2}.

$$\frac{x^{2}}{\left(\sqrt{x^{2}+y^{2}}\right)^{2}}+\frac{y^{2}}{x^{2}+y^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Expand \left(\sqrt{x^{2}+y^{2}}\right)^{2}.

$$\frac{x^{2}}{x^{2}+y^{2}}+\frac{y^{2}}{x^{2}+y^{2}}$$

Since \frac{x^{2}}{x^{2}+y^{2}} and \frac{y^{2}}{x^{2}+y^{2}} have the same denominator, add them by adding their numerators.

$$\frac{x^{2}+y^{2}}{x^{2}+y^{2}}$$

Cancel out x^{2}+y^{2} in both numerator and denominator.

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Factor

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