Question

select all of the solutions to the equation $t^2 = -16$.
$t = 4$
$t = 4i$
$t = -4$
$t = -4i$
$t = 16$
$t = 16i$
$t = -16$
$t = -16i$

Explanation:

Step1: Recall the imaginary unit

The imaginary unit \( i \) is defined as \( i = \sqrt{-1} \), so \( i^2=-1 \).

Step2: Solve the equation \( t^2 = -16 \)

Take the square root of both sides: \( t=\pm\sqrt{-16} \).
Simplify \( \sqrt{-16} \) as \( \sqrt{16\times(-1)}=\sqrt{16}\times\sqrt{-1}=4i \) (since \( \sqrt{ab}=\sqrt{a}\cdot\sqrt{b} \) for non - negative real numbers \( a,b \) and here we extend it to complex numbers with \( \sqrt{-1}=i \)).
So \( t = \pm4i \), that is \( t = 4i \) or \( t=-4i \).

Answer:

\( t = 4i \), \( t=-4i \) (The correct options are: \( t = 4i \), \( t=-4i \))