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Evaluate the following expression:

(iv) [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]


Sagot :

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].
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