Solve (frac{sqrt{46}}{46}-1/3)^2

$$( \frac { \sqrt { 46 } } { 46 } - \frac { 1 } { 3 } ) ^ { 2 }$$

Evaluate

$$-\frac{\sqrt{46}}{69}+\frac{55}{414}\approx 0.034555604$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 46 and 3 is 138. Multiply \frac{\sqrt{46}}{46} times \frac{3}{3}. Multiply \frac{1}{3} times \frac{46}{46}.

$$\left(\frac{3\sqrt{46}}{138}-\frac{46}{138}\right)^{2}$$

Since \frac{3\sqrt{46}}{138} and \frac{46}{138} have the same denominator, subtract them by subtracting their numerators.

$$\left(\frac{3\sqrt{46}-46}{138}\right)^{2}$$

To raise \frac{3\sqrt{46}-46}{138} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(3\sqrt{46}-46\right)^{2}}{138^{2}}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3\sqrt{46}-46\right)^{2}.

$$\frac{9\left(\sqrt{46}\right)^{2}-276\sqrt{46}+2116}{138^{2}}$$

The square of \sqrt{46} is 46.

$$\frac{9\times 46-276\sqrt{46}+2116}{138^{2}}$$

Multiply 9 and 46 to get 414.

$$\frac{414-276\sqrt{46}+2116}{138^{2}}$$

Add 414 and 2116 to get 2530.

$$\frac{2530-276\sqrt{46}}{138^{2}}$$

Calculate 138 to the power of 2 and get 19044.

$$\frac{2530-276\sqrt{46}}{19044}$$

Expand

$$-\frac{\sqrt{46}}{69} + \frac{55}{414} = 0.034555604$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 46 and 3 is 138. Multiply \frac{\sqrt{46}}{46} times \frac{3}{3}. Multiply \frac{1}{3} times \frac{46}{46}.

$$\left(\frac{3\sqrt{46}}{138}-\frac{46}{138}\right)^{2}$$

Since \frac{3\sqrt{46}}{138} and \frac{46}{138} have the same denominator, subtract them by subtracting their numerators.

$$\left(\frac{3\sqrt{46}-46}{138}\right)^{2}$$

To raise \frac{3\sqrt{46}-46}{138} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(3\sqrt{46}-46\right)^{2}}{138^{2}}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3\sqrt{46}-46\right)^{2}.

$$\frac{9\left(\sqrt{46}\right)^{2}-276\sqrt{46}+2116}{138^{2}}$$

The square of \sqrt{46} is 46.

$$\frac{9\times 46-276\sqrt{46}+2116}{138^{2}}$$

Multiply 9 and 46 to get 414.

$$\frac{414-276\sqrt{46}+2116}{138^{2}}$$

Add 414 and 2116 to get 2530.

$$\frac{2530-276\sqrt{46}}{138^{2}}$$

Calculate 138 to the power of 2 and get 19044.

$$\frac{2530-276\sqrt{46}}{19044}$$