To evaluate the expression \( \frac{1}{3}(x + \frac{9}{7}) + 3^4 \) when \( x = \frac{12}{7} \), we will substitute \( x \) with \( \frac{12}{7} \) and simplify the expression step by step.
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Substitute \( x \): \[ \frac{1}{3}\left(\frac{12}{7} + \frac{9}{7}\right) + 3^4 \]
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Combine the fractions: \[ \frac{12}{7} + \frac{9}{7} = \frac{21}{7} = 3 \]
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Substitute this back into the expression: \[ \frac{1}{3}(3) + 3^4 \]
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Simplify \( \frac{1}{3}(3) \): \[ 1 \]
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Calculate \( 3^4 \): \[ 3^4 = 81 \]
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Add the results together: \[ 1 + 81 = 82 \]
So, the value of the expression when \( x = \frac{12}{7} \) is \( \boxed{82} \).
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