Solve 3/12sqrt{5-sqrt{3}}

$$\frac { 3 } { 12 \sqrt { 5 } - \sqrt { 3 } }$$

Evaluate

$$\frac{\sqrt{3}+12\sqrt{5}}{239}\approx 0.11951827$$

Solution Steps

Rationalize the denominator of \frac{3}{12\sqrt{5}-\sqrt{3}} by multiplying numerator and denominator by 12\sqrt{5}+\sqrt{3}.

$$\frac{3\left(12\sqrt{5}+\sqrt{3}\right)}{\left(12\sqrt{5}-\sqrt{3}\right)\left(12\sqrt{5}+\sqrt{3}\right)}$$

Consider \left(12\sqrt{5}-\sqrt{3}\right)\left(12\sqrt{5}+\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{3\left(12\sqrt{5}+\sqrt{3}\right)}{\left(12\sqrt{5}\right)^{2}-\left(\sqrt{3}\right)^{2}}$$

Expand \left(12\sqrt{5}\right)^{2}.

$$\frac{3\left(12\sqrt{5}+\sqrt{3}\right)}{12^{2}\left(\sqrt{5}\right)^{2}-\left(\sqrt{3}\right)^{2}}$$

Calculate 12 to the power of 2 and get 144.

$$\frac{3\left(12\sqrt{5}+\sqrt{3}\right)}{144\left(\sqrt{5}\right)^{2}-\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{3\left(12\sqrt{5}+\sqrt{3}\right)}{144\times 5-\left(\sqrt{3}\right)^{2}}$$

Multiply 144 and 5 to get 720.

$$\frac{3\left(12\sqrt{5}+\sqrt{3}\right)}{720-\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{3\left(12\sqrt{5}+\sqrt{3}\right)}{720-3}$$

Subtract 3 from 720 to get 717.

$$\frac{3\left(12\sqrt{5}+\sqrt{3}\right)}{717}$$

Divide 3\left(12\sqrt{5}+\sqrt{3}\right) by 717 to get \frac{1}{239}\left(12\sqrt{5}+\sqrt{3}\right).

$$\frac{1}{239}\left(12\sqrt{5}+\sqrt{3}\right)$$

Use the distributive property to multiply \frac{1}{239} by 12\sqrt{5}+\sqrt{3}.

$$\frac{1}{239}\times 12\sqrt{5}+\frac{1}{239}\sqrt{3}$$

Multiply \frac{1}{239} and 12 to get \frac{12}{239}.

$$\frac{12}{239}\sqrt{5}+\frac{1}{239}\sqrt{3}$$