Simplify each of the following powers of i. i^15=square RET i -i 1 X-1

To simplify powers of \(i\), we use the fact that \(i\) is the imaginary unit, where \(i = \sqrt{-1}\). The powers of \(i\) cycle every four exponents:<br /><br />\[ i^1 = i \]<br />\[ i^2 = -1 \]<br />\[ i^3 = -i \]<br />\[ i^4 = 1 \]<br />\[ i^5 = i \]<br /><br />This pattern repeats every four powers. To simplify \(i^{15}\), we can find the remainder when 15 is divided by 4:<br /><br />\[ 15 \mod 4 = 3 \]<br /><br />This means \(i^{15}\) is equivalent to \(i^3\):<br /><br />\[ i^{15} = i^3 = -i \]<br /><br />So, the simplified form of \(i^{15}\) is:<br /><br />\[ i^{15} = -i \]