Solve =(3/sqrt{3})^2-2(3/sqrt{3})(5/sqrt{2})+(5/sqrt{2})^2

$$= ( \frac { 3 } { \sqrt { 3 } } ) ^ { 2 } - 2 ( \frac { 3 } { \sqrt { 3 } } ) ( \frac { 5 } { \sqrt { 2 } } ) + ( \frac { 5 } { \sqrt { 2 } } ) ^ { 2 }$$

Evaluate

$$\frac{31}{2}-5\sqrt{6}\approx 3.252551286$$

Solution Steps

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

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

The square of \sqrt{3} is 3.

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

Cancel out 3 and 3.

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

The square of \sqrt{3} is 3.

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

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

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

The square of \sqrt{3} is 3.

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

Cancel out 3 and 3.

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

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

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

The square of \sqrt{2} is 2.

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

Cancel out 2 and 2.

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

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

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

The square of \sqrt{2} is 2.

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

To raise \frac{5\sqrt{2}}{2} to a power, raise both numerator and denominator to the power and then divide.

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

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

$$3-5\sqrt{6}+\frac{\left(5\sqrt{2}\right)^{2}}{2^{2}}$$

Expand \left(5\sqrt{2}\right)^{2}.

$$3-5\sqrt{6}+\frac{5^{2}\left(\sqrt{2}\right)^{2}}{2^{2}}$$

Calculate 5 to the power of 2 and get 25.

$$3-5\sqrt{6}+\frac{25\left(\sqrt{2}\right)^{2}}{2^{2}}$$

The square of \sqrt{2} is 2.

$$3-5\sqrt{6}+\frac{25\times 2}{2^{2}}$$

Multiply 25 and 2 to get 50.

$$3-5\sqrt{6}+\frac{50}{2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$3-5\sqrt{6}+\frac{50}{4}$$

Reduce the fraction \frac{50}{4} to lowest terms by extracting and canceling out 2.

$$3-5\sqrt{6}+\frac{25}{2}$$

Add 3 and \frac{25}{2} to get \frac{31}{2}.

$$\frac{31}{2}-5\sqrt{6}$$