Solve (frac{3sqrt{46}}{23}-2/3)^2

$$( \frac { 3 \sqrt { 46 } } { 23 } - \frac { 2 } { 3 } ) ^ { 2 }$$

Evaluate

$$-\frac{4\sqrt{46}}{23}+\frac{254}{207}\approx 0.047517491$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 23 and 3 is 69. Multiply \frac{3\sqrt{46}}{23} times \frac{3}{3}. Multiply \frac{2}{3} times \frac{23}{23}.

$$\left(\frac{3\times 3\sqrt{46}}{69}-\frac{2\times 23}{69}\right)^{2}$$

Since \frac{3\times 3\sqrt{46}}{69} and \frac{2\times 23}{69} have the same denominator, subtract them by subtracting their numerators.

$$\left(\frac{3\times 3\sqrt{46}-2\times 23}{69}\right)^{2}$$

Do the multiplications in 3\times 3\sqrt{46}-2\times 23.

$$\left(\frac{9\sqrt{46}-46}{69}\right)^{2}$$

To raise \frac{9\sqrt{46}-46}{69} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(9\sqrt{46}-46\right)^{2}}{69^{2}}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(9\sqrt{46}-46\right)^{2}.

$$\frac{81\left(\sqrt{46}\right)^{2}-828\sqrt{46}+2116}{69^{2}}$$

The square of \sqrt{46} is 46.

$$\frac{81\times 46-828\sqrt{46}+2116}{69^{2}}$$

Multiply 81 and 46 to get 3726.

$$\frac{3726-828\sqrt{46}+2116}{69^{2}}$$

Add 3726 and 2116 to get 5842.

$$\frac{5842-828\sqrt{46}}{69^{2}}$$

Calculate 69 to the power of 2 and get 4761.

$$\frac{5842-828\sqrt{46}}{4761}$$

Expand

$$-\frac{4 \sqrt{46}}{23} + \frac{254}{207} = 0.047517491$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 23 and 3 is 69. Multiply \frac{3\sqrt{46}}{23} times \frac{3}{3}. Multiply \frac{2}{3} times \frac{23}{23}.

$$\left(\frac{3\times 3\sqrt{46}}{69}-\frac{2\times 23}{69}\right)^{2}$$

Since \frac{3\times 3\sqrt{46}}{69} and \frac{2\times 23}{69} have the same denominator, subtract them by subtracting their numerators.

$$\left(\frac{3\times 3\sqrt{46}-2\times 23}{69}\right)^{2}$$

Do the multiplications in 3\times 3\sqrt{46}-2\times 23.

$$\left(\frac{9\sqrt{46}-46}{69}\right)^{2}$$

To raise \frac{9\sqrt{46}-46}{69} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(9\sqrt{46}-46\right)^{2}}{69^{2}}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(9\sqrt{46}-46\right)^{2}.

$$\frac{81\left(\sqrt{46}\right)^{2}-828\sqrt{46}+2116}{69^{2}}$$

The square of \sqrt{46} is 46.

$$\frac{81\times 46-828\sqrt{46}+2116}{69^{2}}$$

Multiply 81 and 46 to get 3726.

$$\frac{3726-828\sqrt{46}+2116}{69^{2}}$$

Add 3726 and 2116 to get 5842.

$$\frac{5842-828\sqrt{46}}{69^{2}}$$

Calculate 69 to the power of 2 and get 4761.

$$\frac{5842-828\sqrt{46}}{4761}$$