Solve 1/2*(2+frac{9sqrt{10}}{2})*frac{9sqrt{2}}{2}-1/2*frac{9sqrt{2}}{2}*frac{9sqrt{2}}{2}

$$\frac{ 1 }{ 2 } \times (2+ \frac{ 9 \sqrt{ 10 } }{ 2 } ) \times \frac{ 9 \sqrt{ 2 } }{ 2 } - \frac{ 1 }{ 2 } \times \frac { 9 \sqrt{ 2 } } { 2 } \times \frac { 9 \sqrt{ 2 } } { 2 }$$

Evaluate

$$\frac{9\left(2\sqrt{2}+9\sqrt{5}-9\right)}{4}\approx 31.394337575$$

Short Solution Steps

Multiply \frac{9\sqrt{2}}{2} and \frac{9\sqrt{2}}{2} to get \left(\frac{9\sqrt{2}}{2}\right)^{2}.

$$\frac{1}{2}\left(2+\frac{9\sqrt{10}}{2}\right)\times \frac{9\sqrt{2}}{2}-\frac{1}{2}\times \left(\frac{9\sqrt{2}}{2}\right)^{2}$$

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

$$\frac{1}{2}\left(\frac{2\times 2}{2}+\frac{9\sqrt{10}}{2}\right)\times \frac{9\sqrt{2}}{2}-\frac{1}{2}\times \left(\frac{9\sqrt{2}}{2}\right)^{2}$$

Since \frac{2\times 2}{2} and \frac{9\sqrt{10}}{2} have the same denominator, add them by adding their numerators.

$$\frac{1}{2}\times \frac{2\times 2+9\sqrt{10}}{2}\times \frac{9\sqrt{2}}{2}-\frac{1}{2}\times \left(\frac{9\sqrt{2}}{2}\right)^{2}$$

Do the multiplications in 2\times 2+9\sqrt{10}.

$$\frac{1}{2}\times \frac{4+9\sqrt{10}}{2}\times \frac{9\sqrt{2}}{2}-\frac{1}{2}\times \left(\frac{9\sqrt{2}}{2}\right)^{2}$$

Multiply \frac{1}{2} times \frac{4+9\sqrt{10}}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{4+9\sqrt{10}}{2\times 2}\times \frac{9\sqrt{2}}{2}-\frac{1}{2}\times \left(\frac{9\sqrt{2}}{2}\right)^{2}$$

Multiply \frac{4+9\sqrt{10}}{2\times 2} times \frac{9\sqrt{2}}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(4+9\sqrt{10}\right)\times 9\sqrt{2}}{2\times 2\times 2}-\frac{1}{2}\times \left(\frac{9\sqrt{2}}{2}\right)^{2}$$

To raise \frac{9\sqrt{2}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(4+9\sqrt{10}\right)\times 9\sqrt{2}}{2\times 2\times 2}-\frac{1}{2}\times \frac{\left(9\sqrt{2}\right)^{2}}{2^{2}}$$

Multiply \frac{1}{2} times \frac{\left(9\sqrt{2}\right)^{2}}{2^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(4+9\sqrt{10}\right)\times 9\sqrt{2}}{2\times 2\times 2}-\frac{\left(9\sqrt{2}\right)^{2}}{2\times 2^{2}}$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$\frac{\left(4+9\sqrt{10}\right)\times 9\sqrt{2}}{2\times 2\times 2}-\frac{\left(9\sqrt{2}\right)^{2}}{2^{3}}$$

Expand \left(9\sqrt{2}\right)^{2}.

$$\frac{\left(4+9\sqrt{10}\right)\times 9\sqrt{2}}{2\times 2\times 2}-\frac{9^{2}\left(\sqrt{2}\right)^{2}}{2^{3}}$$

Calculate 9 to the power of 2 and get 81.

$$\frac{\left(4+9\sqrt{10}\right)\times 9\sqrt{2}}{2\times 2\times 2}-\frac{81\left(\sqrt{2}\right)^{2}}{2^{3}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(4+9\sqrt{10}\right)\times 9\sqrt{2}}{2\times 2\times 2}-\frac{81\times 2}{2^{3}}$$

Multiply 81 and 2 to get 162.

$$\frac{\left(4+9\sqrt{10}\right)\times 9\sqrt{2}}{2\times 2\times 2}-\frac{162}{2^{3}}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{\left(4+9\sqrt{10}\right)\times 9\sqrt{2}}{2\times 2\times 2}-\frac{162}{8}$$

Reduce the fraction \frac{162}{8} to lowest terms by extracting and canceling out 2.

$$\frac{\left(4+9\sqrt{10}\right)\times 9\sqrt{2}}{2\times 2\times 2}-\frac{81}{4}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\times 2\times 2 and 4 is 8. Multiply \frac{81}{4} times \frac{2}{2}.

$$\frac{\left(4+9\sqrt{10}\right)\times 9\sqrt{2}}{8}-\frac{81\times 2}{8}$$

Since \frac{\left(4+9\sqrt{10}\right)\times 9\sqrt{2}}{8} and \frac{81\times 2}{8} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(4+9\sqrt{10}\right)\times 9\sqrt{2}-81\times 2}{8}$$

Do the multiplications in \left(4+9\sqrt{10}\right)\times 9\sqrt{2}-81\times 2.

$$\frac{36\sqrt{2}+162\sqrt{5}-162}{8}$$

Factor

$$\frac{9 {(2 \sqrt{2} + 9 \sqrt{5} - 9)}}{4} = 31.394337575049676$$