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2 to the power of 3 over 2

\( \small{ \begin{array}{rcl} 2^{ \frac 32 } \\ &=& 2^{ \frac{1+2}{2} } \\ &=& 2^{ \frac{1}{2}+\frac{2}{2} } \\ &=& 2^{ \frac{1}{2}+ 1 } \\ &=& 2^{ \frac{1}{2} }\cdot 2^1 \\ &=& 2^{ \frac{1}{2} }\cdot 2 \\ &=& \sqrt{2}\cdot 2 \\ &=& 2 \cdot \sqrt{2}\\ &=& 2.82842712475 \end{array} }\)

\(2^{\frac{3}{2}}=2^{1\frac{1}{2}}=2^{1}\times2^{\frac{1}{2}}=2\times\sqrt{2}=\sqrt{2^{2}\times\sqrt{2}^{2}}=\sqrt{4\times2}=\sqrt{8} \mbox{ so } 2^{\frac{3}{2}}=\sqrt{8}\)