Solve frac{[-frac{2sqrt{6}t}{3(t^{2}+2)}]^{2}+8/3(t^2+2)}{(1+t^2)[-4/3(t^2+2)]^2}

$$\frac { [ - \frac { 2 \sqrt { 6 } t } { 3 ( t ^ { 2 } + 2 ) } ] ^ { 2 } + \frac { 8 } { 3 ( t ^ { 2 } + 2 ) } } { ( 1 + t ^ { 2 } ) [ - \frac { 4 } { 3 ( t ^ { 2 } + 2 ) } ] ^ { 2 } }$$

Evaluate

$$3$$

Short Solution Steps

Calculate -\frac{2\sqrt{6}t}{3\left(t^{2}+2\right)} to the power of 2 and get \left(\frac{2\sqrt{6}t}{3\left(t^{2}+2\right)}\right)^{2}.

$$\frac{\left(\frac{2\sqrt{6}t}{3\left(t^{2}+2\right)}\right)^{2}+\frac{8}{3\left(t^{2}+2\right)}}{\left(1+t^{2}\right)\left(-\frac{4}{3\left(t^{2}+2\right)}\right)^{2}}$$

Calculate -\frac{4}{3\left(t^{2}+2\right)} to the power of 2 and get \left(\frac{4}{3\left(t^{2}+2\right)}\right)^{2}.

$$\frac{\left(\frac{2\sqrt{6}t}{3\left(t^{2}+2\right)}\right)^{2}+\frac{8}{3\left(t^{2}+2\right)}}{\left(1+t^{2}\right)\times \left(\frac{4}{3\left(t^{2}+2\right)}\right)^{2}}$$

To raise \frac{2\sqrt{6}t}{3\left(t^{2}+2\right)} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{\left(2\sqrt{6}t\right)^{2}}{\left(3\left(t^{2}+2\right)\right)^{2}}+\frac{8}{3\left(t^{2}+2\right)}}{\left(1+t^{2}\right)\times \left(\frac{4}{3\left(t^{2}+2\right)}\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(3\left(t^{2}+2\right)\right)^{2} and 3\left(t^{2}+2\right) is 9\left(t^{2}+2\right)^{2}. Multiply \frac{8}{3\left(t^{2}+2\right)} times \frac{3\left(t^{2}+2\right)}{3\left(t^{2}+2\right)}.

$$\frac{\frac{\left(2\sqrt{6}t\right)^{2}}{9\left(t^{2}+2\right)^{2}}+\frac{8\times 3\left(t^{2}+2\right)}{9\left(t^{2}+2\right)^{2}}}{\left(1+t^{2}\right)\times \left(\frac{4}{3\left(t^{2}+2\right)}\right)^{2}}$$

Since \frac{\left(2\sqrt{6}t\right)^{2}}{9\left(t^{2}+2\right)^{2}} and \frac{8\times 3\left(t^{2}+2\right)}{9\left(t^{2}+2\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\frac{\left(2\sqrt{6}t\right)^{2}+8\times 3\left(t^{2}+2\right)}{9\left(t^{2}+2\right)^{2}}}{\left(1+t^{2}\right)\times \left(\frac{4}{3\left(t^{2}+2\right)}\right)^{2}}$$

To raise \frac{4}{3\left(t^{2}+2\right)} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{\left(2\sqrt{6}t\right)^{2}+8\times 3\left(t^{2}+2\right)}{9\left(t^{2}+2\right)^{2}}}{\left(1+t^{2}\right)\times \frac{4^{2}}{\left(3\left(t^{2}+2\right)\right)^{2}}}$$

Express \left(1+t^{2}\right)\times \frac{4^{2}}{\left(3\left(t^{2}+2\right)\right)^{2}} as a single fraction.

$$\frac{\frac{\left(2\sqrt{6}t\right)^{2}+8\times 3\left(t^{2}+2\right)}{9\left(t^{2}+2\right)^{2}}}{\frac{\left(1+t^{2}\right)\times 4^{2}}{\left(3\left(t^{2}+2\right)\right)^{2}}}$$

Divide \frac{\left(2\sqrt{6}t\right)^{2}+8\times 3\left(t^{2}+2\right)}{9\left(t^{2}+2\right)^{2}} by \frac{\left(1+t^{2}\right)\times 4^{2}}{\left(3\left(t^{2}+2\right)\right)^{2}} by multiplying \frac{\left(2\sqrt{6}t\right)^{2}+8\times 3\left(t^{2}+2\right)}{9\left(t^{2}+2\right)^{2}} by the reciprocal of \frac{\left(1+t^{2}\right)\times 4^{2}}{\left(3\left(t^{2}+2\right)\right)^{2}}.

$$\frac{\left(\left(2\sqrt{6}t\right)^{2}+8\times 3\left(t^{2}+2\right)\right)\times \left(3\left(t^{2}+2\right)\right)^{2}}{9\left(t^{2}+2\right)^{2}\left(1+t^{2}\right)\times 4^{2}}$$

Factor the expressions that are not already factored.

$$\frac{4\times 12\left(t^{2}+1\right)\times \left(3\left(t^{2}+2\right)\right)^{2}}{9\times 4^{2}\left(t^{2}+1\right)\left(t^{2}+2\right)^{2}}$$

Cancel out 3\times 4\times 4\left(t^{2}+1\right) in both numerator and denominator.

$$\frac{\left(3\left(t^{2}+2\right)\right)^{2}}{3\left(t^{2}+2\right)^{2}}$$

Expand the expression.

$$\frac{9t^{4}+36t^{2}+36}{3t^{4}+12t^{2}+12}$$

Factor the expressions that are not already factored.

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

Cancel out 3\left(t^{2}+2\right)^{2} in both numerator and denominator.

$$3$$

Factor

$$3$$