To find the value of [tex]\( a \)[/tex] that satisfies the equation [tex]\( 9 = \left(\frac{1}{27}\right)^{2 + 3a} \)[/tex], we need to examine each given option for [tex]\( a \)[/tex]. Let's go through each option and check if it holds true for the equation.
### Evaluate for each option:
#### Option 1: [tex]\( a = -\frac{11}{3} \)[/tex]
[tex]\[ 2 + 3a = 2 + 3 \left(-\frac{11}{3}\right) = 2 - 11 = -9 \][/tex]
Now substituting this back in the equation:
[tex]\[ \left(\frac{1}{27}\right)^{-9} = 27^9 \][/tex]
We need to check if [tex]\( 27^9 = 9 \)[/tex]. Clearly, [tex]\( 27^9 \)[/tex] is not equal to 9, so this option is incorrect.
#### Option 2: [tex]\( a = -\frac{7}{3} \)[/tex]
[tex]\[ 2 + 3a = 2 + 3 \left(-\frac{7}{3}\right) = 2 - 7 = -5 \][/tex]
Now substituting this back in the equation:
[tex]\[ \left(\frac{1}{27}\right)^{-5} = 27^5 \][/tex]
We need to check if [tex]\( 27^5 = 9 \)[/tex]. Clearly, [tex]\( 27^5 \)[/tex] is not equal to 9, so this option is incorrect.
#### Option 3: [tex]\( a = \frac{7}{3} \)[/tex]
[tex]\[ 2 + 3a = 2 + 3 \left(\frac{7}{3}\right) = 2 + 7 = 9 \][/tex]
Now substituting this back in the equation:
[tex]\[ \left(\frac{1}{27}\right)^9 = \frac{1}{27^9} \][/tex]
We need to check if [tex]\( \frac{1}{27^9} = 9 \)[/tex]. Clearly, [tex]\( \frac{1}{27^9} \)[/tex] is not equal to 9, so this option is incorrect.
#### Option 4: [tex]\( a = \frac{11}{3} \)[/tex]
[tex]\[ 2 + 3a = 2 + 3 \left(\frac{11}{3}\right) = 2 + 11 = 13 \][/tex]
Now substituting this back in the equation:
[tex]\[ \left(\frac{1}{27}\right)^{13} = \frac{1}{27^{13}} \][/tex]
We need to check if [tex]\( \frac{1}{27^{13}} = 9 \)[/tex]. Clearly, [tex]\( \frac{1}{27^{13}} \)[/tex] is not equal to 9, so this option is incorrect.
### Conclusion:
None of the provided values for [tex]\( a \)[/tex] satisfy the equation [tex]\( 9 = \left(\frac{1}{27}\right)^{2 + 3a} \)[/tex]. As per the detailed checks, there is no correct value for [tex]\( a \)[/tex] among the given choices.
