Solve frac{1/3^1+sqrt{3}}{sqrt{3}}

$$\frac{ \frac{ 1 }{ { 3 }^{ 1 } + \sqrt{ 3 } } }{ \sqrt{ 3 } }$$

Evaluate

$$\frac{\sqrt{3}-1}{6}\approx 0.122008468$$

Solution Steps

Express \frac{\frac{1}{3^{1}+\sqrt{3}}}{\sqrt{3}} as a single fraction.

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

Calculate 3 to the power of 1 and get 3.

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

Rationalize the denominator of \frac{1}{\left(3+\sqrt{3}\right)\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

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

The square of \sqrt{3} is 3.

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

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

$$\frac{\sqrt{3}}{9+3\sqrt{3}}$$

Rationalize the denominator of \frac{\sqrt{3}}{9+3\sqrt{3}} by multiplying numerator and denominator by 9-3\sqrt{3}.

$$\frac{\sqrt{3}\left(9-3\sqrt{3}\right)}{\left(9+3\sqrt{3}\right)\left(9-3\sqrt{3}\right)}$$

Consider \left(9+3\sqrt{3}\right)\left(9-3\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}.

$$\frac{\sqrt{3}\left(9-3\sqrt{3}\right)}{9^{2}-\left(3\sqrt{3}\right)^{2}}$$

Calculate 9 to the power of 2 and get 81.

$$\frac{\sqrt{3}\left(9-3\sqrt{3}\right)}{81-\left(3\sqrt{3}\right)^{2}}$$

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

$$\frac{\sqrt{3}\left(9-3\sqrt{3}\right)}{81-3^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\sqrt{3}\left(9-3\sqrt{3}\right)}{81-9\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\sqrt{3}\left(9-3\sqrt{3}\right)}{81-9\times 3}$$

Multiply 9 and 3 to get 27.

$$\frac{\sqrt{3}\left(9-3\sqrt{3}\right)}{81-27}$$

Subtract 27 from 81 to get 54.

$$\frac{\sqrt{3}\left(9-3\sqrt{3}\right)}{54}$$

Use the distributive property to multiply \sqrt{3} by 9-3\sqrt{3}.

$$\frac{9\sqrt{3}-3\left(\sqrt{3}\right)^{2}}{54}$$

The square of \sqrt{3} is 3.

$$\frac{9\sqrt{3}-3\times 3}{54}$$

Multiply -3 and 3 to get -9.

$$\frac{9\sqrt{3}-9}{54}$$