To solve the expression [tex]\(\frac{\sqrt[4]{32} \cdot \sqrt{4}}{2^{-1}}\)[/tex], we'll break it down into smaller parts and solve each component individually. Here are the detailed steps:
1. Calculate the fourth root of 32:
The fourth root of 32 is denoted as [tex]\(\sqrt[4]{32}\)[/tex]. This is the same as [tex]\(32^{\frac{1}{4}}\)[/tex].
From calculations, we find:
[tex]\[
32^{\frac{1}{4}} \approx 2.3784
\][/tex]
2. Calculate the square root of 4:
The square root of 4 is denoted as [tex]\(\sqrt{4}\)[/tex]. This is the same as [tex]\(4^{\frac{1}{2}}\)[/tex].
From calculations, we find:
[tex]\[
4^{\frac{1}{2}} = 2
\][/tex]
3. Calculate [tex]\(2^{-1}\)[/tex]:
[tex]\(2^{-1}\)[/tex] is the same as [tex]\(\frac{1}{2}\)[/tex]. So:
[tex]\[
2^{-1} = 0.5
\][/tex]
Now, we need to perform the operations within the original expression. Plugging in the values we calculated:
4. Multiply the results of [tex]\(\sqrt[4]{32}\)[/tex] and [tex]\(\sqrt{4}\)[/tex]:
[tex]\[
\sqrt[4]{32} \cdot \sqrt{4} = 2.3784 \cdot 2 = 4.7568
\][/tex]
5. Divide by [tex]\(2^{-1}\)[/tex]:
Since [tex]\(2^{-1} = 0.5\)[/tex], we have:
[tex]\[
\frac{4.7568}{0.5} = 9.5137
\][/tex]
So, the calculated value of the expression [tex]\(\frac{\sqrt[4]{32} \cdot \sqrt{4}}{2^{-1}}\)[/tex] is approximately [tex]\(9.5137\)[/tex].
Thus, the final result is:
[tex]\[
\frac{\sqrt[4]{32} \cdot \sqrt{4}}{2^{-1}} \approx 9.5137
\][/tex]
