Solve =frac{t^{2}+1}{2t}((t^2+1)^2/12t^2+t^2/2+1/2)

$$= \frac { t ^ { 2 } + 1 } { 2 t } ( \frac { ( t ^ { 2 } + 1 ) ^ { 2 } } { 12 t ^ { 2 } } + \frac { t ^ { 2 } } { 2 } + \frac { 1 } { 2 } )$$

Evaluate

$$\frac{\left(7t^{2}+1\right)\left(t^{2}+1\right)^{2}}{24t^{3}}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 12t^{2} and 2 is 12t^{2}. Multiply \frac{t^{2}}{2} times \frac{6t^{2}}{6t^{2}}.

$$\frac{t^{2}+1}{2t}\left(\frac{\left(t^{2}+1\right)^{2}}{12t^{2}}+\frac{t^{2}\times 6t^{2}}{12t^{2}}+\frac{1}{2}\right)$$

Since \frac{\left(t^{2}+1\right)^{2}}{12t^{2}} and \frac{t^{2}\times 6t^{2}}{12t^{2}} have the same denominator, add them by adding their numerators.

$$\frac{t^{2}+1}{2t}\left(\frac{\left(t^{2}+1\right)^{2}+t^{2}\times 6t^{2}}{12t^{2}}+\frac{1}{2}\right)$$

Do the multiplications in \left(t^{2}+1\right)^{2}+t^{2}\times 6t^{2}.

$$\frac{t^{2}+1}{2t}\left(\frac{t^{4}+2t^{2}+1+6t^{4}}{12t^{2}}+\frac{1}{2}\right)$$

Combine like terms in t^{4}+2t^{2}+1+6t^{4}.

$$\frac{t^{2}+1}{2t}\left(\frac{7t^{4}+2t^{2}+1}{12t^{2}}+\frac{1}{2}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 12t^{2} and 2 is 12t^{2}. Multiply \frac{1}{2} times \frac{6t^{2}}{6t^{2}}.

$$\frac{t^{2}+1}{2t}\left(\frac{7t^{4}+2t^{2}+1}{12t^{2}}+\frac{6t^{2}}{12t^{2}}\right)$$

Since \frac{7t^{4}+2t^{2}+1}{12t^{2}} and \frac{6t^{2}}{12t^{2}} have the same denominator, add them by adding their numerators.

$$\frac{t^{2}+1}{2t}\times \frac{7t^{4}+2t^{2}+1+6t^{2}}{12t^{2}}$$

Combine like terms in 7t^{4}+2t^{2}+1+6t^{2}.

$$\frac{t^{2}+1}{2t}\times \frac{7t^{4}+8t^{2}+1}{12t^{2}}$$

Multiply \frac{t^{2}+1}{2t} times \frac{7t^{4}+8t^{2}+1}{12t^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(t^{2}+1\right)\left(7t^{4}+8t^{2}+1\right)}{2t\times 12t^{2}}$$

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

$$\frac{\left(t^{2}+1\right)\left(7t^{4}+8t^{2}+1\right)}{2t^{3}\times 12}$$

Multiply 2 and 12 to get 24.

$$\frac{\left(t^{2}+1\right)\left(7t^{4}+8t^{2}+1\right)}{24t^{3}}$$

Use the distributive property to multiply t^{2}+1 by 7t^{4}+8t^{2}+1 and combine like terms.

$$\frac{7t^{6}+15t^{4}+9t^{2}+1}{24t^{3}}$$

Expand

$$\frac{7t^{3}}{24}+\frac{5t}{8}+\frac{3}{8t}+\frac{1}{24t^{3}}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 12t^{2} and 2 is 12t^{2}. Multiply \frac{t^{2}}{2} times \frac{6t^{2}}{6t^{2}}.

$$\frac{t^{2}+1}{2t}\left(\frac{\left(t^{2}+1\right)^{2}}{12t^{2}}+\frac{t^{2}\times 6t^{2}}{12t^{2}}+\frac{1}{2}\right)$$

Since \frac{\left(t^{2}+1\right)^{2}}{12t^{2}} and \frac{t^{2}\times 6t^{2}}{12t^{2}} have the same denominator, add them by adding their numerators.

$$\frac{t^{2}+1}{2t}\left(\frac{\left(t^{2}+1\right)^{2}+t^{2}\times 6t^{2}}{12t^{2}}+\frac{1}{2}\right)$$

Do the multiplications in \left(t^{2}+1\right)^{2}+t^{2}\times 6t^{2}.

$$\frac{t^{2}+1}{2t}\left(\frac{t^{4}+2t^{2}+1+6t^{4}}{12t^{2}}+\frac{1}{2}\right)$$

Combine like terms in t^{4}+2t^{2}+1+6t^{4}.

$$\frac{t^{2}+1}{2t}\left(\frac{7t^{4}+2t^{2}+1}{12t^{2}}+\frac{1}{2}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 12t^{2} and 2 is 12t^{2}. Multiply \frac{1}{2} times \frac{6t^{2}}{6t^{2}}.

$$\frac{t^{2}+1}{2t}\left(\frac{7t^{4}+2t^{2}+1}{12t^{2}}+\frac{6t^{2}}{12t^{2}}\right)$$

Since \frac{7t^{4}+2t^{2}+1}{12t^{2}} and \frac{6t^{2}}{12t^{2}} have the same denominator, add them by adding their numerators.

$$\frac{t^{2}+1}{2t}\times \frac{7t^{4}+2t^{2}+1+6t^{2}}{12t^{2}}$$

Combine like terms in 7t^{4}+2t^{2}+1+6t^{2}.

$$\frac{t^{2}+1}{2t}\times \frac{7t^{4}+8t^{2}+1}{12t^{2}}$$

Multiply \frac{t^{2}+1}{2t} times \frac{7t^{4}+8t^{2}+1}{12t^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(t^{2}+1\right)\left(7t^{4}+8t^{2}+1\right)}{2t\times 12t^{2}}$$

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

$$\frac{\left(t^{2}+1\right)\left(7t^{4}+8t^{2}+1\right)}{2t^{3}\times 12}$$

Multiply 2 and 12 to get 24.

$$\frac{\left(t^{2}+1\right)\left(7t^{4}+8t^{2}+1\right)}{24t^{3}}$$

Use the distributive property to multiply t^{2}+1 by 7t^{4}+8t^{2}+1 and combine like terms.

$$\frac{7t^{6}+15t^{4}+9t^{2}+1}{24t^{3}}$$