Which of the following expression has the largest value?

Options

A. `(3^4)^(13)`
B. `[(3^(30))^(12)]^(1/10)`
C. `3^(30) + 3^(30) + 3^(30)`
D. `4(3^(51))`

Answer

To determine which of the given expressions has the largest value, let's evaluate each expression step by step.

### Given Expressions:
1. \( 3^4 \) raised to the power of 13
2. \( (3^{30})^{12} \) raised to the power of \( \frac{1}{10} \)
3. \( 3^{30} + 3^{30} + 3^{30} \)
4. \( 4 \times 3^{51} \)

### Step 1: Evaluate Expression A
Expression A is \( (3^4)^{13} \).

Using the power of a power property:
\[ (3^4)^{13} = 3^{4 \times 13} = 3^{52} \]

### Step 2: Evaluate Expression B
Expression B is \( (3^{30})^{12} \) raised to the power of \( \frac{1}{10} \).

First, simplify the inner expression:
\[ (3^{30})^{12} = 3^{30 \times 12} = 3^{360} \]
Now raise this to the power of \( \frac{1}{10} \):
\[ (3^{360})^{\frac{1}{10}} = 3^{360 \times \frac{1}{10}} = 3^{36} \]

### Step 3: Evaluate Expression C
Expression C is \( 3^{30} + 3^{30} + 3^{30} \).

Since there are three identical terms, we can factor it:
\[ 3^{30} + 3^{30} + 3^{30} = 3 \times 3^{30} = 3^1 \times 3^{30} = 3^{1 + 30} = 3^{31} \]

### Step 4: Evaluate Expression D
Expression D is \( 4 \times 3^{51} \).

This expression remains as it is for now.

### Step 5: Compare All Expressions
Now we have:
- Expression A: \( 3^{52} \)
- Expression B: \( 3^{36} \)
- Expression C: \( 3^{31} \)
- Expression D: \( 4 \times 3^{51} \)

To compare \( 4 \times 3^{51} \) with \( 3^{52} \):
\[ 4 \times 3^{51} = 3^{51} \times 4 = 3^{51} \times 3^{1.26186} \quad (\text{since } 4 \approx 3^{1.26186}) \]
This means \( 4 \times 3^{51} \) is less than \( 3^{52} \).

### Conclusion
Now we can see:
- \( 3^{52} > 3^{36} \)
- \( 3^{52} > 3^{31} \)
- \( 3^{52} > 4 \times 3^{51} \)

Thus, the expression with the largest value is:
\[ \text{Option A: } 3^{52} \]

### Final Answer
The largest expression is \( 3^{52} \).

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