Question

simplify.
$i^7$
$i^7 = \square$

Explanation:

Step1: Recall the powers of \(i\)

We know that \(i = \sqrt{-1}\), \(i^2=-1\), \(i^3 = i^2\times i=-1\times i = -i\), \(i^4=(i^2)^2=(-1)^2 = 1\). The powers of \(i\) repeat every 4. So we can express \(i^7\) in terms of these known powers.

Step2: Rewrite \(i^7\)

We can write \(i^7\) as \(i^{4 + 3}\). Using the property of exponents \(a^{m + n}=a^m\times a^n\), we have \(i^{4+3}=i^4\times i^3\).

Step3: Substitute the known values

We know \(i^4 = 1\) and \(i^3=-i\). So \(i^4\times i^3=1\times(-i)=-i\).

Answer:

\(-i\)