Solve (1/sqrt{x+sqrt{y}})x^2x^2*1/sqrt{x-sqrt{y}}

$$( \frac{ 1 }{ \sqrt{ x } + \sqrt{ y } } ) { x }^{ 2 } { x }^{ 2 } \times \frac{ 1 }{ \sqrt{ x } - \sqrt{ y } }$$

Evaluate

$$\frac{x^{4}}{x-y}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$\frac{1}{\sqrt{x}+\sqrt{y}}x^{4}\times \frac{1}{\sqrt{x}-\sqrt{y}}$$

Express \frac{1}{\sqrt{x}+\sqrt{y}}x^{4} as a single fraction.

$$\frac{x^{4}}{\sqrt{x}+\sqrt{y}}\times \frac{1}{\sqrt{x}-\sqrt{y}}$$

Multiply \frac{x^{4}}{\sqrt{x}+\sqrt{y}} times \frac{1}{\sqrt{x}-\sqrt{y}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{x^{4}}{\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)}$$

Consider \left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{x^{4}}{\left(\sqrt{x}\right)^{2}-\left(\sqrt{y}\right)^{2}}$$

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

$$\frac{x^{4}}{x-\left(\sqrt{y}\right)^{2}}$$

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

$$\frac{x^{4}}{x-y}$$

Factor

$$\frac{x^{4}}{x-y}$$