To begin simplifying the expression
[tex]\[
\left(\frac{\left|2 r^2 t\right|^3}{4 t^2}\right)^2,
\][/tex]
we should first focus on the absolute value inside the numerator, [tex]\( |2 r^2 t| \)[/tex].
The absolute value function ensures that the expression inside it is non-negative, regardless of the values of [tex]\( r \)[/tex] and [tex]\( t \)[/tex].
Since [tex]\( 2 r^2 t \)[/tex] is a product of constants and variables, taking the absolute value does not affect the variable's powers or coefficients; it simply makes sure the overall term is non-negative. Therefore,
[tex]\[
|2 r^2 t| = 2 |r^2| |t| = 2 r^2 |t| (since r^2 \geq 0, its absolute value doesn't change it).
\][/tex]
Once the absolute value is simplified, we can then cube this term as follows:
[tex]\[
(2 r^2 t)^3.
\][/tex]
Cubing the term involves raising both the coefficient and the variables inside to the power of three:
[tex]\[
(2 r^2 t)^3 = 2^3 \cdot (r^2)^3 \cdot (t)^3 = 8 r^6 t^3.
\][/tex]
Hence, the result of raising the absolute value and cubing inside the numerator is:
[tex]\[
(2 r^2 t)^3 = 8 r^6 t^3.
\][/tex]
