Solve frac{(frac{sqrt{3}}{1})^{2}+4*(1/sqrt{2})^2+3*(2/sqrt{3})^2+5*0^2}{2+2-(sqrt{3})^2}

$$\frac { ( \frac { \sqrt { 3 } } { 1 } ) ^ { 2 } + 4 \times ( \frac { 1 } { \sqrt { 2 } } ) ^ { 2 } + 3 \times ( \frac { 2 } { \sqrt { 3 } } ) ^ { 2 } + 5 \times 0 ^ { 2 } } { 2 + 2 - ( \sqrt { 3 } ) ^ { 2 } }$$

Evaluate

$$9$$

Solution Steps

Anything divided by one gives itself.

$$\frac{\left(\sqrt{3}\right)^{2}+4\times \left(\frac{1}{\sqrt{2}}\right)^{2}+3\times \left(\frac{2}{\sqrt{3}}\right)^{2}+5\times 0^{2}}{2+2-\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{3+4\times \left(\frac{1}{\sqrt{2}}\right)^{2}+3\times \left(\frac{2}{\sqrt{3}}\right)^{2}+5\times 0^{2}}{2+2-\left(\sqrt{3}\right)^{2}}$$

Rationalize the denominator of \frac{1}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{3+4\times \left(\frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}\right)^{2}+3\times \left(\frac{2}{\sqrt{3}}\right)^{2}+5\times 0^{2}}{2+2-\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{3+4\times \left(\frac{\sqrt{2}}{2}\right)^{2}+3\times \left(\frac{2}{\sqrt{3}}\right)^{2}+5\times 0^{2}}{2+2-\left(\sqrt{3}\right)^{2}}$$

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

$$\frac{3+4\times \frac{\left(\sqrt{2}\right)^{2}}{2^{2}}+3\times \left(\frac{2}{\sqrt{3}}\right)^{2}+5\times 0^{2}}{2+2-\left(\sqrt{3}\right)^{2}}$$

Express 4\times \frac{\left(\sqrt{2}\right)^{2}}{2^{2}} as a single fraction.

$$\frac{3+\frac{4\left(\sqrt{2}\right)^{2}}{2^{2}}+3\times \left(\frac{2}{\sqrt{3}}\right)^{2}+5\times 0^{2}}{2+2-\left(\sqrt{3}\right)^{2}}$$

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

$$\frac{3+\frac{4\left(\sqrt{2}\right)^{2}}{2^{2}}+3\times \left(\frac{2\sqrt{3}}{\left(\sqrt{3}\right)^{2}}\right)^{2}+5\times 0^{2}}{2+2-\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{3+\frac{4\left(\sqrt{2}\right)^{2}}{2^{2}}+3\times \left(\frac{2\sqrt{3}}{3}\right)^{2}+5\times 0^{2}}{2+2-\left(\sqrt{3}\right)^{2}}$$

To raise \frac{2\sqrt{3}}{3} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{3+\frac{4\left(\sqrt{2}\right)^{2}}{2^{2}}+3\times \frac{\left(2\sqrt{3}\right)^{2}}{3^{2}}+5\times 0^{2}}{2+2-\left(\sqrt{3}\right)^{2}}$$

Express 3\times \frac{\left(2\sqrt{3}\right)^{2}}{3^{2}} as a single fraction.

$$\frac{3+\frac{4\left(\sqrt{2}\right)^{2}}{2^{2}}+\frac{3\times \left(2\sqrt{3}\right)^{2}}{3^{2}}+5\times 0^{2}}{2+2-\left(\sqrt{3}\right)^{2}}$$

Cancel out 3 in both numerator and denominator.

$$\frac{3+\frac{4\left(\sqrt{2}\right)^{2}}{2^{2}}+\frac{\left(2\sqrt{3}\right)^{2}}{3}+5\times 0^{2}}{2+2-\left(\sqrt{3}\right)^{2}}$$

Calculate 0 to the power of 2 and get 0.

$$\frac{3+\frac{4\left(\sqrt{2}\right)^{2}}{2^{2}}+\frac{\left(2\sqrt{3}\right)^{2}}{3}+5\times 0}{2+2-\left(\sqrt{3}\right)^{2}}$$

Multiply 5 and 0 to get 0.

$$\frac{3+\frac{4\left(\sqrt{2}\right)^{2}}{2^{2}}+\frac{\left(2\sqrt{3}\right)^{2}}{3}+0}{2+2-\left(\sqrt{3}\right)^{2}}$$

Add 3 and 0 to get 3.

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

Expand \left(2\sqrt{3}\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{3} is 3.

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

Multiply 4 and 3 to get 12.

$$\frac{3+\frac{4\left(\sqrt{2}\right)^{2}}{2^{2}}+\frac{12}{3}}{2+2-\left(\sqrt{3}\right)^{2}}$$

Divide 12 by 3 to get 4.

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

Add 3 and 4 to get 7.

$$\frac{7+\frac{4\left(\sqrt{2}\right)^{2}}{2^{2}}}{2+2-\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{7+\frac{4\times 2}{2^{2}}}{2+2-\left(\sqrt{3}\right)^{2}}$$

Multiply 4 and 2 to get 8.

$$\frac{7+\frac{8}{2^{2}}}{2+2-\left(\sqrt{3}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{7+\frac{8}{4}}{2+2-\left(\sqrt{3}\right)^{2}}$$

Divide 8 by 4 to get 2.

$$\frac{7+2}{2+2-\left(\sqrt{3}\right)^{2}}$$

Add 7 and 2 to get 9.

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

Add 2 and 2 to get 4.

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

The square of \sqrt{3} is 3.

$$\frac{9}{4-3}$$

Subtract 3 from 4 to get 1.

$$\frac{9}{1}$$

Anything divided by one gives itself.

$$9$$

Factor

$$3^{2}$$