Solve 1/sqrt{5-4}+1/sqrt{5+4}

$$\frac { 1 } { \sqrt { 5 } - 4 } + \frac { 1 } { \sqrt { 5 } + 4 }$$

Evaluate

$$-\frac{2\sqrt{5}}{11}\approx -0.406557814$$

Short Solution Steps

Rationalize the denominator of \frac{1}{\sqrt{5}-4} by multiplying numerator and denominator by \sqrt{5}+4.

$$\frac{\sqrt{5}+4}{\left(\sqrt{5}-4\right)\left(\sqrt{5}+4\right)}+\frac{1}{\sqrt{5}+4}$$

Consider \left(\sqrt{5}-4\right)\left(\sqrt{5}+4\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\sqrt{5}+4}{\left(\sqrt{5}\right)^{2}-4^{2}}+\frac{1}{\sqrt{5}+4}$$

Square \sqrt{5}. Square 4.

$$\frac{\sqrt{5}+4}{5-16}+\frac{1}{\sqrt{5}+4}$$

Subtract 16 from 5 to get -11.

$$\frac{\sqrt{5}+4}{-11}+\frac{1}{\sqrt{5}+4}$$

Multiply both numerator and denominator by -1.

$$\frac{-\sqrt{5}-4}{11}+\frac{1}{\sqrt{5}+4}$$

Rationalize the denominator of \frac{1}{\sqrt{5}+4} by multiplying numerator and denominator by \sqrt{5}-4.

$$\frac{-\sqrt{5}-4}{11}+\frac{\sqrt{5}-4}{\left(\sqrt{5}+4\right)\left(\sqrt{5}-4\right)}$$

Consider \left(\sqrt{5}+4\right)\left(\sqrt{5}-4\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{-\sqrt{5}-4}{11}+\frac{\sqrt{5}-4}{\left(\sqrt{5}\right)^{2}-4^{2}}$$

Square \sqrt{5}. Square 4.

$$\frac{-\sqrt{5}-4}{11}+\frac{\sqrt{5}-4}{5-16}$$

Subtract 16 from 5 to get -11.

$$\frac{-\sqrt{5}-4}{11}+\frac{\sqrt{5}-4}{-11}$$

Multiply both numerator and denominator by -1.

$$\frac{-\sqrt{5}-4}{11}+\frac{-\sqrt{5}+4}{11}$$

Since \frac{-\sqrt{5}-4}{11} and \frac{-\sqrt{5}+4}{11} have the same denominator, add them by adding their numerators.

$$\frac{-\sqrt{5}-4-\sqrt{5}+4}{11}$$

Do the calculations in -\sqrt{5}-4-\sqrt{5}+4.

$$\frac{-2\sqrt{5}}{11}$$