Answer
$$\kappa \left( t \right) = \frac{6}{{\left| t \right|{{\left( {4 + 9{t^2}} \right)}^{3/2}}}}$$
Work Step by Step
$$\eqalign{
& {\bf{r}}\left( t \right) = {t^2}{\bf{i}} + {t^3}{\bf{j}} \cr
& {\text{Calculate the derivatives }}{\bf{r}}'\left( t \right){\text{ and }}{\bf{r}}{\text{''}}\left( t \right) \cr
& {\bf{r}}'\left( t \right) = \frac{d}{{dt}}\left[ {{t^2}} \right]{\bf{i}} + \frac{d}{{dt}}\left[ {{t^3}} \right]{\bf{j}} \cr
& {\bf{r}}'\left( t \right) = 2t{\bf{i}} + 3{t^2}{\bf{j}} \cr
& and \cr
& {\bf{r}}''\left( t \right) = \frac{d}{{dt}}\left[ {2t} \right]{\bf{i}} + \frac{d}{{dt}}\left[ {3{t^2}} \right]{\bf{j}} \cr
& {\bf{r}}''\left( t \right) = 2{\bf{i}} + 6t{\bf{j}} \cr
& \cr
& {\text{Calculate the cross product }}{\bf{r}}'\left( t \right) \times {\bf{r}}''\left( t \right) \cr} $$
\[\begin{gathered}
{\mathbf{r}}'\left( t \right) \times {\mathbf{r}}''\left( t \right) = \left| {\begin{array}{*{20}{c}}
{\mathbf{i}}&{\mathbf{j}}&{\mathbf{k}} \\
{2t}&{3{t^2}}&0 \\
2&{6t}&0
\end{array}} \right| \hfill \\
{\mathbf{r}}'\left( t \right) \times {\mathbf{r}}''\left( t \right) = \left| {\begin{array}{*{20}{c}}
{3{t^2}}&0 \\
{6t}&0
\end{array}} \right|{\mathbf{i}} - \left| {\begin{array}{*{20}{c}}
{2t}&0 \\
2&0
\end{array}} \right|{\mathbf{j}} + \left| {\begin{array}{*{20}{c}}
{2t}&{3{t^2}} \\
2&{6t}
\end{array}} \right|{\mathbf{k}} \hfill \\
{\mathbf{r}}'\left( t \right) \times {\mathbf{r}}''\left( t \right) = 0{\mathbf{i}} - 0{\mathbf{j}} + \left( {12{t^2} - 6{t^2}} \right){\mathbf{k}} \hfill \\
{\mathbf{r}}'\left( t \right) \times {\mathbf{r}}''\left( t \right) = 6{t^2}{\mathbf{k}} \hfill \\
\end{gathered} \]
$$\eqalign{
& {\text{Use the formula }}\left( 3 \right)\,\,\,\kappa \left( t \right) = \frac{{\left\| {{\bf{r}}'\left( t \right) \times {\bf{r}}''\left( t \right)} \right\|}}{{{{\left\| {{\bf{r}}'\left( t \right)} \right\|}^3}}}.{\text{ Thus}}{\text{,}} \cr
& \kappa \left( t \right) = \frac{{\left\| {6{t^2}{\bf{k}}} \right\|}}{{{{\left\| {2t{\bf{i}} + 3{t^2}{\bf{j}}} \right\|}^3}}} \cr
& \kappa \left( t \right) = \frac{{6{t^2}}}{{{{\left( {\sqrt {{{\left( {2t} \right)}^2} + {{\left( {3{t^2}} \right)}^2}} } \right)}^3}}} \cr
& \kappa \left( t \right) = \frac{{6{t^2}}}{{{{\left( {\sqrt {4{t^2} + 9{t^4}} } \right)}^3}}} \cr
& \kappa \left( t \right) = \frac{{6{t^2}}}{{{{\left( {\left| t \right|\sqrt {4 + 9{t^2}} } \right)}^3}}} \cr
& \kappa \left( t \right) = \frac{6}{{\left| t \right|{{\left( {4 + 9{t^2}} \right)}^{3/2}}}} \cr} $$
