Solve (2/sqrt{19+sqrt{17}}+2/sqrt{17+sqrt{15}}+sqrt{15})-sqrt{19}

$$( \frac { 2 } { \sqrt { 19 } + \sqrt { 17 } } + \frac { 2 } { \sqrt { 17 } + \sqrt { 15 } } + \sqrt { 15 } ) - \sqrt { 19 }$$

Evaluate

$$0$$

Solution Steps

Rationalize the denominator of \frac{2}{\sqrt{19}+\sqrt{17}} by multiplying numerator and denominator by \sqrt{19}-\sqrt{17}.

$$\frac{2\left(\sqrt{19}-\sqrt{17}\right)}{\left(\sqrt{19}+\sqrt{17}\right)\left(\sqrt{19}-\sqrt{17}\right)}+\frac{2}{\sqrt{17}+\sqrt{15}}+\sqrt{15}-\sqrt{19}$$

Consider \left(\sqrt{19}+\sqrt{17}\right)\left(\sqrt{19}-\sqrt{17}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{2\left(\sqrt{19}-\sqrt{17}\right)}{\left(\sqrt{19}\right)^{2}-\left(\sqrt{17}\right)^{2}}+\frac{2}{\sqrt{17}+\sqrt{15}}+\sqrt{15}-\sqrt{19}$$

Square \sqrt{19}. Square \sqrt{17}.

$$\frac{2\left(\sqrt{19}-\sqrt{17}\right)}{19-17}+\frac{2}{\sqrt{17}+\sqrt{15}}+\sqrt{15}-\sqrt{19}$$

Subtract 17 from 19 to get 2.

$$\frac{2\left(\sqrt{19}-\sqrt{17}\right)}{2}+\frac{2}{\sqrt{17}+\sqrt{15}}+\sqrt{15}-\sqrt{19}$$

Cancel out 2 and 2.

$$\sqrt{19}-\sqrt{17}+\frac{2}{\sqrt{17}+\sqrt{15}}+\sqrt{15}-\sqrt{19}$$

Rationalize the denominator of \frac{2}{\sqrt{17}+\sqrt{15}} by multiplying numerator and denominator by \sqrt{17}-\sqrt{15}.

$$\sqrt{19}-\sqrt{17}+\frac{2\left(\sqrt{17}-\sqrt{15}\right)}{\left(\sqrt{17}+\sqrt{15}\right)\left(\sqrt{17}-\sqrt{15}\right)}+\sqrt{15}-\sqrt{19}$$

Consider \left(\sqrt{17}+\sqrt{15}\right)\left(\sqrt{17}-\sqrt{15}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\sqrt{19}-\sqrt{17}+\frac{2\left(\sqrt{17}-\sqrt{15}\right)}{\left(\sqrt{17}\right)^{2}-\left(\sqrt{15}\right)^{2}}+\sqrt{15}-\sqrt{19}$$

Square \sqrt{17}. Square \sqrt{15}.

$$\sqrt{19}-\sqrt{17}+\frac{2\left(\sqrt{17}-\sqrt{15}\right)}{17-15}+\sqrt{15}-\sqrt{19}$$

Subtract 15 from 17 to get 2.

$$\sqrt{19}-\sqrt{17}+\frac{2\left(\sqrt{17}-\sqrt{15}\right)}{2}+\sqrt{15}-\sqrt{19}$$

Cancel out 2 and 2.

$$\sqrt{19}-\sqrt{17}+\sqrt{17}-\sqrt{15}+\sqrt{15}-\sqrt{19}$$

Combine -\sqrt{17} and \sqrt{17} to get 0.

$$\sqrt{19}-\sqrt{15}+\sqrt{15}-\sqrt{19}$$

Combine -\sqrt{15} and \sqrt{15} to get 0.

$$\sqrt{19}-\sqrt{19}$$

Combine \sqrt{19} and -\sqrt{19} to get 0.

$$0$$

Factor

$$0$$