Chapter 10 - Exponents and Radicals - 10.8 The Complex Numbers - 10.8 Exercise Set - Page 687: 96

$i$

Using $i^2=-1$ and the laws of exponents, the given expression, $ 5i^5+4i^3 ,$ is equivalent to \begin{array}{l}\require{cancel} 5i^4\cdot i+4i^2\cdot i \\\\= 5(i^2)^2\cdot i+4i^2\cdot i \\\\= 5(-1)^2\cdot i+4(-1)\cdot i \\\\= 5(1)\cdot i+4(-1)\cdot i \\\\= 5i-4i \\\\= i \end{array}