Solve (1/sqrt{3}-1)^2+91^2/sqrt{3}-1

$$( \frac { 1 } { \sqrt { 3 } } - 1 ) ^ { 2 } + 9 \frac { 1 ^ { 2 } } { \sqrt { 3 } } - 1$$

Evaluate

$$\frac{7\sqrt{3}+1}{3}\approx 4.374785218$$

Short Solution Steps

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

$$\left(\frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}-1\right)^{2}+9\times \frac{1^{2}}{\sqrt{3}}-1$$

The square of \sqrt{3} is 3.

$$\left(\frac{\sqrt{3}}{3}-1\right)^{2}+9\times \frac{1^{2}}{\sqrt{3}}-1$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{3}{3}.

$$\left(\frac{\sqrt{3}}{3}-\frac{3}{3}\right)^{2}+9\times \frac{1^{2}}{\sqrt{3}}-1$$

Since \frac{\sqrt{3}}{3} and \frac{3}{3} have the same denominator, subtract them by subtracting their numerators.

$$\left(\frac{\sqrt{3}-3}{3}\right)^{2}+9\times \frac{1^{2}}{\sqrt{3}}-1$$

To raise \frac{\sqrt{3}-3}{3} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(\sqrt{3}-3\right)^{2}}{3^{2}}+9\times \frac{1^{2}}{\sqrt{3}}-1$$

Calculate 1 to the power of 2 and get 1.

$$\frac{\left(\sqrt{3}-3\right)^{2}}{3^{2}}+9\times \frac{1}{\sqrt{3}}-1$$

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

$$\frac{\left(\sqrt{3}-3\right)^{2}}{3^{2}}+9\times \frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}-1$$

The square of \sqrt{3} is 3.

$$\frac{\left(\sqrt{3}-3\right)^{2}}{3^{2}}+9\times \frac{\sqrt{3}}{3}-1$$

Cancel out 3, the greatest common factor in 9 and 3.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 3\sqrt{3}-1 times \frac{3^{2}}{3^{2}}.

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

Since \frac{\left(\sqrt{3}-3\right)^{2}}{3^{2}} and \frac{\left(3\sqrt{3}-1\right)\times 3^{2}}{3^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(\sqrt{3}-3\right)^{2}+\left(3\sqrt{3}-1\right)\times 3^{2}.

$$\frac{\left(\sqrt{3}\right)^{2}-6\sqrt{3}+9+27\sqrt{3}-9}{3^{2}}$$

Do the calculations in \left(\sqrt{3}\right)^{2}-6\sqrt{3}+9+27\sqrt{3}-9.

$$\frac{3+21\sqrt{3}}{3^{2}}$$

Expand 3^{2}.

$$\frac{3+21\sqrt{3}}{9}$$