Solve 1/sqrt{3+5}-1/sqrt{3-5}

$$\frac { 1 } { \sqrt { 3 } + 5 } - \frac { 1 } { \sqrt { 3 } - 5 }$$

Evaluate

$$\frac{5}{11}\approx 0.454545455$$

Short Solution Steps

Rationalize the denominator of \frac{1}{\sqrt{3}+5} by multiplying numerator and denominator by \sqrt{3}-5.

$$\frac{\sqrt{3}-5}{\left(\sqrt{3}+5\right)\left(\sqrt{3}-5\right)}-\frac{1}{\sqrt{3}-5}$$

Consider \left(\sqrt{3}+5\right)\left(\sqrt{3}-5\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\sqrt{3}-5}{\left(\sqrt{3}\right)^{2}-5^{2}}-\frac{1}{\sqrt{3}-5}$$

Square \sqrt{3}. Square 5.

$$\frac{\sqrt{3}-5}{3-25}-\frac{1}{\sqrt{3}-5}$$

Subtract 25 from 3 to get -22.

$$\frac{\sqrt{3}-5}{-22}-\frac{1}{\sqrt{3}-5}$$

Multiply both numerator and denominator by -1.

$$\frac{-\sqrt{3}+5}{22}-\frac{1}{\sqrt{3}-5}$$

Rationalize the denominator of \frac{1}{\sqrt{3}-5} by multiplying numerator and denominator by \sqrt{3}+5.

$$\frac{-\sqrt{3}+5}{22}-\frac{\sqrt{3}+5}{\left(\sqrt{3}-5\right)\left(\sqrt{3}+5\right)}$$

Consider \left(\sqrt{3}-5\right)\left(\sqrt{3}+5\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{-\sqrt{3}+5}{22}-\frac{\sqrt{3}+5}{\left(\sqrt{3}\right)^{2}-5^{2}}$$

Square \sqrt{3}. Square 5.

$$\frac{-\sqrt{3}+5}{22}-\frac{\sqrt{3}+5}{3-25}$$

Subtract 25 from 3 to get -22.

$$\frac{-\sqrt{3}+5}{22}-\frac{\sqrt{3}+5}{-22}$$

Multiply both numerator and denominator by -1.

$$\frac{-\sqrt{3}+5}{22}-\frac{-\sqrt{3}-5}{22}$$

Since \frac{-\sqrt{3}+5}{22} and \frac{-\sqrt{3}-5}{22} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-\sqrt{3}+5-\left(-\sqrt{3}-5\right)}{22}$$

Do the multiplications in -\sqrt{3}+5-\left(-\sqrt{3}-5\right).

$$\frac{-\sqrt{3}+5+\sqrt{3}+5}{22}$$

Do the calculations in -\sqrt{3}+5+\sqrt{3}+5.

$$\frac{10}{22}$$

Reduce the fraction \frac{10}{22} to lowest terms by extracting and canceling out 2.

$$\frac{5}{11}$$

Factor

$$\frac{5}{11} = 0.45454545454545453$$