Solve 17.15/2sqrt{5+sqrt{3}}

$$\frac { 17.15 } { 2 \sqrt { 5 } + \sqrt { 3 } }$$

Evaluate

$$\frac{343\sqrt{5}}{170}-\frac{343\sqrt{3}}{340}\approx 2.764262369$$

Solution Steps

Rationalize the denominator of \frac{17.15}{2\sqrt{5}+\sqrt{3}} by multiplying numerator and denominator by 2\sqrt{5}-\sqrt{3}.

$$\frac{17.15\left(2\sqrt{5}-\sqrt{3}\right)}{\left(2\sqrt{5}+\sqrt{3}\right)\left(2\sqrt{5}-\sqrt{3}\right)}$$

Consider \left(2\sqrt{5}+\sqrt{3}\right)\left(2\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}.

$$\frac{17.15\left(2\sqrt{5}-\sqrt{3}\right)}{\left(2\sqrt{5}\right)^{2}-\left(\sqrt{3}\right)^{2}}$$

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

$$\frac{17.15\left(2\sqrt{5}-\sqrt{3}\right)}{2^{2}\left(\sqrt{5}\right)^{2}-\left(\sqrt{3}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{17.15\left(2\sqrt{5}-\sqrt{3}\right)}{4\left(\sqrt{5}\right)^{2}-\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{17.15\left(2\sqrt{5}-\sqrt{3}\right)}{4\times 5-\left(\sqrt{3}\right)^{2}}$$

Multiply 4 and 5 to get 20.

$$\frac{17.15\left(2\sqrt{5}-\sqrt{3}\right)}{20-\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{17.15\left(2\sqrt{5}-\sqrt{3}\right)}{20-3}$$

Subtract 3 from 20 to get 17.

$$\frac{17.15\left(2\sqrt{5}-\sqrt{3}\right)}{17}$$

Divide 17.15\left(2\sqrt{5}-\sqrt{3}\right) by 17 to get \frac{343}{340}\left(2\sqrt{5}-\sqrt{3}\right).

$$\frac{343}{340}\left(2\sqrt{5}-\sqrt{3}\right)$$

Use the distributive property to multiply \frac{343}{340} by 2\sqrt{5}-\sqrt{3}.

$$\frac{343}{170}\sqrt{5}-\frac{343}{340}\sqrt{3}$$