Solve frac{sqrt{3}}{5sqrt{3}+sqrt{2}}

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

Evaluate

$$\frac{15-\sqrt{6}}{73}\approx 0.171924798$$

Solution Steps

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

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

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

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

Expand \left(5\sqrt{3}\right)^{2}.

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

Calculate 5 to the power of 2 and get 25.

$$\frac{\sqrt{3}\left(5\sqrt{3}-\sqrt{2}\right)}{25\left(\sqrt{3}\right)^{2}-\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\sqrt{3}\left(5\sqrt{3}-\sqrt{2}\right)}{25\times 3-\left(\sqrt{2}\right)^{2}}$$

Multiply 25 and 3 to get 75.

$$\frac{\sqrt{3}\left(5\sqrt{3}-\sqrt{2}\right)}{75-\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{3}\left(5\sqrt{3}-\sqrt{2}\right)}{75-2}$$

Subtract 2 from 75 to get 73.

$$\frac{\sqrt{3}\left(5\sqrt{3}-\sqrt{2}\right)}{73}$$

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

$$\frac{5\left(\sqrt{3}\right)^{2}-\sqrt{3}\sqrt{2}}{73}$$

The square of \sqrt{3} is 3.

$$\frac{5\times 3-\sqrt{3}\sqrt{2}}{73}$$

Multiply 5 and 3 to get 15.

$$\frac{15-\sqrt{3}\sqrt{2}}{73}$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{15-\sqrt{6}}{73}$$