Find the value of `3^((4)/(log_(2)9))+27^((1)/(log_(49)9))+81^((1)/(log_(4)3))`

Options

A. 890
B. 860
C. 857
D. none of these

Answer

To solve the expression \( 3^{\frac{4}{\log_{2}9}} + 27^{\frac{1}{\log_{49}9}} + 81^{\frac{1}{\log_{4}3}} \), we will use properties of logarithms. Let's break it down step by step.

### Step 1: Simplify \( 3^{\frac{4}{\log_{2}9}} \)

Using the property of logarithms that states \( \log_{a}b = \frac{1}{\log_{b}a} \), we can rewrite \( \frac{1}{\log_{2}9} \) as \( \log_{9}2 \).

Thus, we have:
\[ 3^{\frac{4}{\log_{2}9}} = 3^{4 \cdot \log_{9}2} = 3^{\log_{9}2^4} = 3^{\log_{9}16} \]
Using the property \( a^{\log_{b}c} = c^{\log_{b}a} \), we can rewrite this as:
\[ 3^{\log_{9}16} = 16^{\log_{9}3} \]

### Step 2: Simplify \( 27^{\frac{1}{\log_{49}9}} \)

Similarly, we can rewrite \( \frac{1}{\log_{49}9} \) as \( \log_{9}49 \).

Thus, we have:
\[ 27^{\frac{1}{\log_{49}9}} = 27^{\log_{9}49} \]
Using the property \( a^{\log_{b}c} = c^{\log_{b}a} \), we can rewrite this as:
\[ 27^{\log_{9}49} = 49^{\log_{9}27} \]

### Step 3: Simplify \( 81^{\frac{1}{\log_{4}3}} \)

Again, we can rewrite \( \frac{1}{\log_{4}3} \) as \( \log_{3}4 \).

Thus, we have:
\[ 81^{\frac{1}{\log_{4}3}} = 81^{\log_{3}4} \]
Using the property \( a^{\log_{b}c} = c^{\log_{b}a} \), we can rewrite this as:
\[ 81^{\log_{3}4} = 4^{\log_{3}81} \]

### Step 4: Combine the results

Now we can combine all the results:
\[ 3^{\frac{4}{\log_{2}9}} + 27^{\frac{1}{\log_{49}9}} + 81^{\frac{1}{\log_{4}3}} = 16^{\log_{9}3} + 49^{\log_{9}27} + 4^{\log_{3}81} \]

### Step 5: Calculate each term

1. **Calculate \( 16^{\log_{9}3} \)**:
- \( 16 = 2^4 \), so \( 16^{\log_{9}3} = (2^4)^{\log_{9}3} = 2^{4\log_{9}3} = 2^{\log_{9}3^4} = 2^{\log_{9}81} = 81^{\log_{9}2} \).

2. **Calculate \( 49^{\log_{9}27} \)**:
- \( 49 = 7^2 \), so \( 49^{\log_{9}27} = (7^2)^{\log_{9}27} = 7^{2\log_{9}27} = 27^{\log_{9}49} \).

3. **Calculate \( 4^{\log_{3}81} \)**:
- \( 4 = 2^2 \), so \( 4^{\log_{3}81} = (2^2)^{\log_{3}81} = 2^{2\log_{3}81} = 81^{\log_{3}4} \).

### Final Calculation

Now we can sum the results:
\[ 16^{\log_{9}3} + 49^{\log_{9}27} + 4^{\log_{3}81} = 256 + 343 + 81 = 680 \]

### Final Answer

Thus, the final value of the expression is:
\[ \boxed{680} \]