Solve 31/sqrt{3+5sqrt{2}}

$$3 \frac { 1 } { \sqrt { 3 } + 5 \sqrt { 2 } }$$

Evaluate

$$\frac{15\sqrt{2}-3\sqrt{3}}{47}\approx 0.340788319$$

Solution Steps

Rationalize the denominator of \frac{1}{\sqrt{3}+5\sqrt{2}} by multiplying numerator and denominator by \sqrt{3}-5\sqrt{2}.

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

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

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

The square of \sqrt{3} is 3.

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

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

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

Calculate 5 to the power of 2 and get 25.

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

The square of \sqrt{2} is 2.

$$3\times \frac{\sqrt{3}-5\sqrt{2}}{3-25\times 2}$$

Multiply 25 and 2 to get 50.

$$3\times \frac{\sqrt{3}-5\sqrt{2}}{3-50}$$

Subtract 50 from 3 to get -47.

$$3\times \frac{\sqrt{3}-5\sqrt{2}}{-47}$$

Multiply both numerator and denominator by -1.

$$3\times \frac{-\sqrt{3}+5\sqrt{2}}{47}$$

Express 3\times \frac{-\sqrt{3}+5\sqrt{2}}{47} as a single fraction.

$$\frac{3\left(-\sqrt{3}+5\sqrt{2}\right)}{47}$$

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

$$\frac{-3\sqrt{3}+15\sqrt{2}}{47}$$