https://file.aiquickdraw.com/mathbot/file/1726722372739hezazlfo.jpg

The image contains two logarithmic expressions to be calculated under the heading "Hitung," which means "Calculate" in Indonesian:

1. \( \text{a.} \ (9)^{\sqrt[3]{\log_2 3}} \)
2. \( \text{b.} \ (8)^{4 \log_{27} 4} \)

Let’s solve each one step by step.

### Part (a): \( (9)^{\sqrt[3]{\log_2 3}} \)

1. Start by expressing 9 as a power of 3: 
   \[
   9 = 3^2
   \]
   So the expression becomes:
   \[
   (3^2)^{\sqrt[3]{\log_2 3}} = 3^{2\sqrt[3]{\log_2 3}}
   \]
2. The result is simplified, but it can remain in this form for now. This expression is difficult to simplify further without numerical approximation.

### Part (b): \( (8)^{4 \log_{27} 4} \)

1. Express 8 and 27 in terms of powers of 2 and 3:
   \[
   8 = 2^3, \quad 27 = 3^3
   \]
   So the expression becomes:
   \[
   (2^3)^{4 \log_{3^3} 4} = 2^{12 \log_3 4 / 3}
   \]
2. Use the property of logarithms: \( \log_{a^b} c = \frac{1}{b} \log_a c \), which simplifies the exponent:
   \[
   2^{4 \log_3 4}
   \]
3. Finally, apply the logarithmic identity \( a^{\log_b c} = c^{\log_b a} \) to simplify the expression:
   \[
   2^{4 \log_3 4} = 4^{\log_3 2^4} = 4^{\log_3 16}
   \]

Let me know if you would like further clarification or a more detailed explanation!

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Here are 5 related questions for practice:

1. What is the value of \( \log_2 16 \)?
2. Simplify \( (27)^{\log_3 3} \).
3. Calculate \( (4)^{\log_2 16} \).
4. How do you simplify \( a^{\log_b c} \) using logarithmic properties?
5. Evaluate \( (25)^{\log_5 5} \).

**Tip:** Always remember logarithmic properties such as \( a^{\log_b c} = c^{\log_b a} \), as they are useful for simplifying complex expressions.

(9)^{\sqrt[3]{\log_2 3}} and (8)^{4 \log_{27} 4}

Learn how to solve logarithmic expressions like (9)^{\sqrt[3]{\log_2 3}} and (8)^{4 \log_{27} 4}. This problem covers logarithmic properties, exponent rules, and algebraic manipulation, suitable for Grades 10-12.

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