Certainly! Let's carefully evaluate the given expression step-by-step:
[tex]\[ \frac{\sqrt{2}}{3} \left(\sqrt{0} + 54 + \sqrt{6}\right) \][/tex]
First, let's calculate each term individually:
1. Calculate [tex]\(\sqrt{2}\)[/tex]:
[tex]\[ \sqrt{2} \approx 1.4142135623730951 \][/tex]
2. Calculate [tex]\(\sqrt{0}\)[/tex]:
[tex]\[ \sqrt{0} = 0.0 \][/tex]
3. The constant term [tex]\(54\)[/tex]:
[tex]\[ 54 \][/tex]
4. Calculate [tex]\(\sqrt{6}\)[/tex]:
[tex]\[ \sqrt{6} \approx 2.449489742783178 \][/tex]
Next, we sum the terms inside the parentheses:
[tex]\[ \sqrt{0} + 54 + \sqrt{6} = 0.0 + 54 + 2.449489742783178 \][/tex]
So,
[tex]\[ 0.0 + 54 + \sqrt{6} = 54 + 2.449489742783178 = 56.44948974278318 \][/tex]
Now, we multiply the denominator by 3:
[tex]\[ 3 \times 56.44948974278318 = 169.34846922834953 \][/tex]
Finally, we divide [tex]\(\sqrt{2}\)[/tex] by this product:
[tex]\[ \frac{\sqrt{2}}{169.34846922834953} \][/tex]
So,
[tex]\[ \frac{1.4142135623730951}{169.34846922834953} \approx 0.008350908448225587 \][/tex]
Thus, the evaluated result of the given expression is approximately:
[tex]\[ \boxed{0.008350908448225587} \][/tex]
