Solve 9sqrt{75}+sqrt{21/2}*3/2sqrt{22/3}

$$9 \sqrt { 75 } + \sqrt { 2 \frac { 1 } { 2 } } \times \frac { 3 } { 2 } \sqrt { 2 \frac { 2 } { 3 } }$$

Evaluate

$$\sqrt{15}+45\sqrt{3}\approx 81.815269687$$

Short Solution Steps

Factor 75=5^{2}\times 3. Rewrite the square root of the product \sqrt{5^{2}\times 3} as the product of square roots \sqrt{5^{2}}\sqrt{3}. Take the square root of 5^{2}.

$$9\times 5\sqrt{3}+\sqrt{\frac{2\times 2+1}{2}}\times \frac{3}{2}\sqrt{\frac{2\times 3+2}{3}}$$

Multiply 9 and 5 to get 45.

$$45\sqrt{3}+\sqrt{\frac{2\times 2+1}{2}}\times \frac{3}{2}\sqrt{\frac{2\times 3+2}{3}}$$

Multiply 2 and 2 to get 4.

$$45\sqrt{3}+\sqrt{\frac{4+1}{2}}\times \frac{3}{2}\sqrt{\frac{2\times 3+2}{3}}$$

Add 4 and 1 to get 5.

$$45\sqrt{3}+\sqrt{\frac{5}{2}}\times \frac{3}{2}\sqrt{\frac{2\times 3+2}{3}}$$

Rewrite the square root of the division \sqrt{\frac{5}{2}} as the division of square roots \frac{\sqrt{5}}{\sqrt{2}}.

$$45\sqrt{3}+\frac{\sqrt{5}}{\sqrt{2}}\times \frac{3}{2}\sqrt{\frac{2\times 3+2}{3}}$$

Rationalize the denominator of \frac{\sqrt{5}}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$45\sqrt{3}+\frac{\sqrt{5}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}\times \frac{3}{2}\sqrt{\frac{2\times 3+2}{3}}$$

The square of \sqrt{2} is 2.

$$45\sqrt{3}+\frac{\sqrt{5}\sqrt{2}}{2}\times \frac{3}{2}\sqrt{\frac{2\times 3+2}{3}}$$

To multiply \sqrt{5} and \sqrt{2}, multiply the numbers under the square root.

$$45\sqrt{3}+\frac{\sqrt{10}}{2}\times \frac{3}{2}\sqrt{\frac{2\times 3+2}{3}}$$

Multiply 2 and 3 to get 6.

$$45\sqrt{3}+\frac{\sqrt{10}}{2}\times \frac{3}{2}\sqrt{\frac{6+2}{3}}$$

Add 6 and 2 to get 8.

$$45\sqrt{3}+\frac{\sqrt{10}}{2}\times \frac{3}{2}\sqrt{\frac{8}{3}}$$

Rewrite the square root of the division \sqrt{\frac{8}{3}} as the division of square roots \frac{\sqrt{8}}{\sqrt{3}}.

$$45\sqrt{3}+\frac{\sqrt{10}}{2}\times \frac{3}{2}\times \frac{\sqrt{8}}{\sqrt{3}}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$45\sqrt{3}+\frac{\sqrt{10}}{2}\times \frac{3}{2}\times \frac{2\sqrt{2}}{\sqrt{3}}$$

Rationalize the denominator of \frac{2\sqrt{2}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$45\sqrt{3}+\frac{\sqrt{10}}{2}\times \frac{3}{2}\times \frac{2\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$45\sqrt{3}+\frac{\sqrt{10}}{2}\times \frac{3}{2}\times \frac{2\sqrt{2}\sqrt{3}}{3}$$

To multiply \sqrt{2} and \sqrt{3}, multiply the numbers under the square root.

$$45\sqrt{3}+\frac{\sqrt{10}}{2}\times \frac{3}{2}\times \frac{2\sqrt{6}}{3}$$

Multiply \frac{\sqrt{10}}{2} times \frac{3}{2} by multiplying numerator times numerator and denominator times denominator.

$$45\sqrt{3}+\frac{\sqrt{10}\times 3}{2\times 2}\times \frac{2\sqrt{6}}{3}$$

Multiply \frac{\sqrt{10}\times 3}{2\times 2} times \frac{2\sqrt{6}}{3} by multiplying numerator times numerator and denominator times denominator.

$$45\sqrt{3}+\frac{\sqrt{10}\times 3\times 2\sqrt{6}}{2\times 2\times 3}$$

Cancel out 2\times 3 in both numerator and denominator.

$$45\sqrt{3}+\frac{\sqrt{6}\sqrt{10}}{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 45\sqrt{3} times \frac{2}{2}.

$$\frac{2\times 45\sqrt{3}}{2}+\frac{\sqrt{6}\sqrt{10}}{2}$$

Since \frac{2\times 45\sqrt{3}}{2} and \frac{\sqrt{6}\sqrt{10}}{2} have the same denominator, add them by adding their numerators.

$$\frac{2\times 45\sqrt{3}+\sqrt{6}\sqrt{10}}{2}$$

Do the multiplications in 2\times 45\sqrt{3}+\sqrt{6}\sqrt{10}.

$$\frac{90\sqrt{3}+2\sqrt{15}}{2}$$

Divide each term of 90\sqrt{3}+2\sqrt{15} by 2 to get 45\sqrt{3}+\sqrt{15}.

$$45\sqrt{3}+\sqrt{15}$$

Factor

$$\sqrt{15} + 45 \sqrt{3} = 81.815269687$$