## College Algebra (10th Edition)

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$\bf{\text{Solution Outline:}}$ To simplify the given expression, $i^6+i^4+i^2+1 ,$ use the laws of exponents and the equivalence $i^2=-1.$ $\bf{\text{Solution Details:}}$ Using the Power Rule of the laws of exponents which is given by $\left( x^m \right)^p=x^{mp},$ the expression above is equivalent to \begin{array}{l}\require{cancel} i^{2\cdot3}+i^{2\cdot2}+i^2+1 \\\\= (i^2)^3+(i^2)^2+i^2+1 .\end{array} Since $i^2=-1,$ the expression above is equivalent to \begin{array}{l}\require{cancel} (-1)^3+(-1)^2+(-1)+1 \\\\= -1+1-1+1 \\\\= 0 .\end{array}