To solve the problem defined by the expression [tex]\(\left[\frac{\sqrt{b}}{a}\right]=a^2 \times \sqrt[4]{b}\)[/tex], follow these steps:
1. Identify the variables and numerical values:
- [tex]\(a = 4\)[/tex]
- [tex]\(b = 9\)[/tex]
2. Compute [tex]\(\sqrt{b}\)[/tex]:
- [tex]\(\sqrt{b} = \sqrt{9} = 3\)[/tex]
3. Compute the fourth root of [tex]\(b\)[/tex]:
- [tex]\(\sqrt[4]{b} = \sqrt[4]{9} = (9)^{1/4}\)[/tex]
4. Calculate [tex]\(\left[\frac{\sqrt{b}}{a}\right]\)[/tex] inside the brackets:
- Place the previously computed value of [tex]\(\sqrt{b}\)[/tex]:
[tex]\[
\left[\frac{3}{4}\right]
\][/tex]
- Evaluate the expression inside the brackets but note that due to its construction, the actual calculation focuses on:
[tex]\[
a^2 \times \sqrt[4]{b}
\][/tex]
5. Compute [tex]\(a^2\)[/tex]:
- [tex]\(a^2 = 4^2 = 16\)[/tex]
6. Combine the results:
- Multiply [tex]\(a^2\)[/tex] by [tex]\(\sqrt[4]{b}\)[/tex]:
- [tex]\(\sqrt[4]{9} = \)[/tex] the fourth root of 9 can be computed. For simplicity, assume this value impacts the overall multiplication correctly.
7. Final Calculation:
Given the components:
[tex]\[
16 \times (9)^{1/4}
\][/tex]
This results in approximately:
[tex]\[
16 \times 1.732... \approx 27.7128
\][/tex]
8. Compare the calculated result with the provided choices:
- The calculated value is approximately [tex]\(27.7128\)[/tex]
So, comparing this value to the provided multiple-choice options, none of the choices [tex]\(70, 72, 60, 62, 65\)[/tex] match the computed value of approximately [tex]\(27.7128\)[/tex].
Thus, the final outcome is:
- [tex]\[
27.7128
\][/tex]
- And there are no appropriate choices that match this value. Hence, the result includes an empty set for feasible choices.
