x^3cdot x^2 12(x^2)^3 (x^3)^2 3x^3+6x

Let's go through each expression step by step:<br /><br />1. \( x^{3} \cdot x^{2} \)<br /><br />When multiplying expressions with the same base, you add the exponents:<br />\[ x^{3} \cdot x^{2} = x^{3+2} = x^{5} \]<br /><br />2. \( 12(x^{2})^{3} \)<br /><br />First, apply the power rule \((a^m)^n = a^{mn}\):<br />\[ (x^{2})^{3} = x^{2 \cdot 3} = x^{6} \]<br /><br />Then multiply by 12:<br />\[ 12(x^{2})^{3} = 12x^{6} \]<br /><br />3. \( (x^{3})^{2} \)<br /><br />Apply the power rule \((a^m)^n = a^{mn}\):<br />\[ (x^{3})^{2} = x^{3 \cdot 2} = x^{6} \]<br /><br />4. \( 3x^{3} + 6x \)<br /><br />This expression is already simplified and does not require any further steps.<br /><br />So, the simplified forms of the given expressions are:<br />1. \( x^{5} \)<br />2. \( 12x^{6} \)<br />3. \( x^{6} \)<br />4. \( 3x^{3} + 6x \)