Let's simplify the given expression step-by-step and evaluate it:
The given expression is:
[tex]$\left(\frac{t^2 r s^2}{4 t^{-2} r^2 s}\right)^0$[/tex]
First, let's simplify the expression inside the parentheses:
1. Combine the terms in the numerator and the denominator:
[tex]\[
\frac{t^2 r s^2}{4 t^{-2} r^2 s}
\][/tex]
2. Simplify each component:
- For [tex]\(t\)[/tex]: [tex]\[
\frac{t^2}{t^{-2}} = t^2 \cdot t^2 = t^{2+2} = t^4
\][/tex]
- For [tex]\(r\)[/tex]: [tex]\[
\frac{r}{r^2} = r^{1-2} = r^{-1} = \frac{1}{r}
\][/tex]
- For [tex]\(s\)[/tex]: [tex]\[
\frac{s^2}{s} = s^{2-1} = s
\][/tex]
3. Simplify the fraction combining all the simplified terms:
[tex]\[
\frac{t^4 \cdot s}{4r}
\][/tex]
Therefore, the simplified form inside the parentheses is:
[tex]\[
\frac{t^4 \cdot s}{4r}
\][/tex]
4. Considering the exponent 0:
Any non-zero expression raised to the power of 0 is equal to 1. Hence,
[tex]\[
\left(\frac{t^4 s}{4 r}\right)^0 = 1
\][/tex]
Therefore, the simplified result of the expression [tex]\(\left(\frac{t^2 r s^2}{4 t^{-2} r^2 s}\right)^0\)[/tex] is:
[tex]\[
\boxed{1}
\][/tex]
