To solve the expression [tex]\(\left[ \frac{116}{9} - \frac{3}{2} \times 16^{-\frac{3}{11}} \right]^{\frac{t}{3}}\)[/tex], we will break it into several steps:
1. Calculate [tex]\(\frac{116}{9}\)[/tex]:
- [tex]\[\frac{116}{9} \approx 12.88888888888889\][/tex]
2. Evaluate [tex]\(16^{-\frac{3}{11}}\)[/tex]:
- Calculate the exponent first:
- [tex]\[-\frac{3}{11}\][/tex]
- The value of [tex]\(16^{-\frac{3}{11}}\)[/tex] can be found using the properties of exponents, particularly involving roots and negatives, but for now, let's treat it as a numerical value:
- [tex]\[16^{-\frac{3}{11}} \approx 0.46946501605734455\][/tex]
3. Multiply [tex]\(\frac{3}{2}\)[/tex] by this result:
- [tex]\[\frac{3}{2} \times 16^{-\frac{3}{11}} \approx \frac{3}{2} \times 0.46946501605734455 \approx 0.7041981829962798\][/tex]
4. Subtract the second term from the first term:
- [tex]\[ \frac{116}{9} - \frac{3}{2} \times 16^{-\frac{3}{11}} \approx 12.88888888888889 - 0.7041981829962798 \approx 12.18469070589261\][/tex]
5. Raise the result to the power of [tex]\(\frac{t}{3}\)[/tex]:
- Assuming [tex]\( t = 3 \)[/tex] for simplicity:
- [tex]\[\left(12.18469070589261\right)^{\frac{3}{3}} = (12.18469070589261)^{1} \approx 12.18469070589261\][/tex]
Therefore, the simplified and calculated value of the expression [tex]\(\left[ \frac{116}{9} - \frac{3}{2} \times 16^{-\frac{3}{11}} \right]^{\frac{t}{3}}\)[/tex] is approximately [tex]\(12.18469070589261\)[/tex] when [tex]\( t = 3 \)[/tex].
To summarize the steps:
1. Calculated [tex]\(\frac{116}{9}\)[/tex] to get approximately 12.88888888888889.
2. Evaluated [tex]\(16^{-\frac{3}{11}}\)[/tex] to be approximately 0.46946501605734455.
3. Multiplied [tex]\(\frac{3}{2}\)[/tex] by 0.46946501605734455 to get approximately 0.7041981829962798.
4. Subtracted this result from 12.88888888888889 to get approximately 12.18469070589261.
5. Raised this result to the power of [tex]\(\frac{t}{3}\)[/tex] (assuming [tex]\(t = 3\)[/tex]) to get approximately 12.18469070589261.
Thus, the value of the expression is around 12.18469070589261.
