Discover answers to your questions with Westonci.ca, the leading Q&A platform that connects you with knowledgeable experts. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
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.
### 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.
Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.