Let's evaluate the given expression step-by-step.
Given:
[tex]\[
\left(\frac{15}{4}\right)^{-1} \times\left(\left(\frac{12}{15}\right)^{-1} \div\left(\frac{144}{225}\right)^{-1}\right)
\][/tex]
### Step 1: Calculate the inverse of each fraction.
1. The inverse of [tex]\(\frac{15}{4}\)[/tex] is [tex]\(\left(\frac{15}{4}\right)^{-1}\)[/tex].
[tex]\[
\left(\frac{15}{4}\right)^{-1} = \frac{4}{15} \approx 0.2667
\][/tex]
2. The inverse of [tex]\(\frac{12}{15}\)[/tex] is [tex]\(\left(\frac{12}{15}\right)^{-1}\)[/tex].
[tex]\[
\left(\frac{12}{15}\right)^{-1} = \frac{15}{12} = 1.25
\][/tex]
3. The inverse of [tex]\(\frac{144}{225}\)[/tex] is [tex]\(\left(\frac{144}{225}\right)^{-1}\)[/tex].
[tex]\[
\left(\frac{144}{225}\right)^{-1} = \frac{225}{144} \approx 1.5625
\][/tex]
### Step 2: Perform the division inside the parentheses.
We need to divide the inverse of [tex]\(\frac{12}{15}\)[/tex] by the inverse of [tex]\(\frac{144}{225}\)[/tex].
[tex]\[
\frac{15}{12} \div \frac{225}{144} = \frac{15}{12} \times \frac{144}{225}
\][/tex]
Simplify the fraction:
[tex]\[
= \frac{15 \times 144}{12 \times 225} = \frac{2160}{2700} = \frac{8}{10} = 0.8
\][/tex]
### Step 3: Multiply with the inverse of [tex]\(\frac{15}{4}\)[/tex].
[tex]\[
\left(\frac{15}{4}\right)^{-1} \times\left(\left(\frac{12}{15}\right)^{-1} \div\left(\frac{144}{225}\right)^{-1}\right)
\][/tex]
[tex]\[
= \frac{4}{15} \times 0.8 = 0.2667 \times 0.8 = 0.2133
\][/tex]
### Final result:
[tex]\[
0.2133
\][/tex]
Thus, the evaluation of the given expression is approximately [tex]\(0.2133\)[/tex].
