To solve the problem [tex]\(\left(\sqrt{\frac{225}{729}} - \sqrt{\frac{25}{144}}\right) \div \sqrt{\frac{16}{81}}\)[/tex], we will break it into a series of manageable steps.
### Step 1: Simplify the expressions inside the square roots
1. [tex]\(\frac{225}{729}\)[/tex]
2. [tex]\(\frac{25}{144}\)[/tex]
3. [tex]\(\frac{16}{81}\)[/tex]
From the problem, we have the values:
- [tex]\(\frac{225}{729} \approx 0.30864197530864196\)[/tex]
- [tex]\(\frac{25}{144} \approx 0.1736111111111111\)[/tex]
- [tex]\(\frac{16}{81} \approx 0.19753086419753085\)[/tex]
### Step 2: Calculate the square roots of these expressions
We now take the square root of the simplified values step by step:
1. [tex]\(\sqrt{0.30864197530864196} \approx 0.5555555555555556\)[/tex]
2. [tex]\(\sqrt{0.1736111111111111} \approx 0.4166666666666667\)[/tex]
3. [tex]\(\sqrt{0.19753086419753085} \approx 0.4444444444444444\)[/tex]
### Step 3: Perform the subtraction and division
First, we subtract the square roots obtained:
[tex]\[
0.5555555555555556 - 0.4166666666666667 \approx 0.1388888888888889
\][/tex]
Next, we divide this result by the square root of the third value:
[tex]\[
\frac{0.1388888888888889}{0.4444444444444444} \approx 0.31250000000000006
\][/tex]
### Conclusion
The value of [tex]\(\left(\sqrt{\frac{225}{729}} - \sqrt{\frac{25}{144}}\right) \div \sqrt{\frac{16}{81}}\)[/tex] is approximately [tex]\(0.3125\)[/tex].
Checking our options, the different values in the choices are fractions. Converting [tex]\(0.3125\)[/tex] to a fraction:
[tex]\[
0.3125 = \frac{5}{16}
\][/tex]
None of the given options:
1) [tex]\(\frac{1}{48}\)[/tex]
2) [tex]\(\frac{5}{48}\)[/tex]
3) [tex]\(\frac{1}{1}\)[/tex]
match [tex]\(0.3125\)[/tex] or [tex]\(\frac{5}{16}\)[/tex]. This indicates that none of the provided choices are correct based on our calculation.
However, in proper context, if this were a question with answer choices, we could recheck if needed, but with given calculations, correct choice isn't provided in the given options.
