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

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

Evaluate

$$\frac{9\sqrt{3}-3}{26}\approx 0.484171433$$

Solution Steps

Calculate 3 to the power of -1 and get \frac{1}{3}.

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

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

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

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

$$\frac{\frac{1}{3}-\sqrt{3}}{\left(\frac{1}{3}\right)^{2}-\left(\sqrt{3}\right)^{2}}$$

Square \frac{1}{3}. Square \sqrt{3}.

$$\frac{\frac{1}{3}-\sqrt{3}}{\frac{1}{9}-3}$$

Subtract 3 from \frac{1}{9} to get -\frac{26}{9}.

$$\frac{\frac{1}{3}-\sqrt{3}}{-\frac{26}{9}}$$

Divide \frac{1}{3}-\sqrt{3} by -\frac{26}{9} by multiplying \frac{1}{3}-\sqrt{3} by the reciprocal of -\frac{26}{9}.

$$\frac{\left(\frac{1}{3}-\sqrt{3}\right)\times 9}{-26}$$

Use the distributive property to multiply \frac{1}{3}-\sqrt{3} by 9.

$$\frac{3-9\sqrt{3}}{-26}$$

Factor

$$\frac{3 {(3 \sqrt{3} - 1)}}{26} = 0.48417143338922675$$