Let's go through the given expression step by step.
The expression we need to evaluate is:
[tex]$(81)^{-1} \times 3^{-5} \times 3^9 \times (64)^{5/6} \times (\sqrt[3]{3})^6$[/tex]
1. Calculate [tex]\((81)^{-1}\)[/tex]:
[tex]$81 = 3^4 \quad \text{(since } 81 = 3 \times 3 \times 3 \times 3\text{)}$[/tex]
Therefore,
[tex]$(81)^{-1} = (3^4)^{-1} = 3^{-4} \approx 0.012345679012345678$[/tex]
2. Calculate [tex]\(3^{-5}\)[/tex]:
[tex]$3^{-5} = \frac{1}{3^5} = \frac{1}{243} \approx 0.00411522633744856$[/tex]
3. Calculate [tex]\(3^9\)[/tex]:
[tex]$3^9 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 19683$[/tex]
4. Calculate [tex]\((64)^{5/6}\)[/tex]:
[tex]$64 = 2^6$[/tex]
Hence,
[tex]$(64)^{5/6} = \left(2^6\right)^{5/6} = 2^{6 \times \frac{5}{6}} = 2^5 = 32 \approx 32.00000000000001$[/tex]
5. Calculate [tex]\((\sqrt[3]{3})^6\)[/tex]:
[tex]$\sqrt[3]{3} = 3^{1/3}$[/tex]
Therefore,
[tex]$(\sqrt[3]{3})^6 = \left(3^{1/3}\right)^6 = 3^{(1/3) \times 6} = 3^2 = 9 \approx 8.999999999999996$[/tex]
Now, we multiply all these terms together:
[tex]$(81)^{-1} \times 3^{-5} \times 3^9 \times (64)^{5/6} \times (\sqrt[3]{3})^6$[/tex]
[tex]$\approx 0.012345679012345678 \times 0.00411522633744856 \times 19683 \times 32.00000000000001 \times 8.999999999999996$[/tex]
After multiplying these values, we get the final result:
[tex]$\approx 287.99999999999994$[/tex]
Hence, the expression evaluates to approximately [tex]\(288\)[/tex].
