Greatest number among the `2^(50), 3^(40), 4^(30)` and `5^(20)` is:

Options

A. `3^(40)`
B. `4^(30)`
C. `5^(20)`
D. `2^(50)`

Answer

To find the greatest number among \(2^{50}\), \(3^{40}\), \(4^{30}\), and \(5^{20}\), we can express all the numbers in terms of a common base.

### Step 1: Rewrite each number in terms of base 2.
1. \(2^{50}\) remains as \(2^{50}\).
2. \(3^{40}\) can be rewritten using logarithms or approximations, but let's keep it as is for now.
3. \(4^{30} = (2^2)^{30} = 2^{60}\).
4. \(5^{20}\) can also be expressed in terms of base 2 using logarithms, but let's keep it as is for now.

### Step 2: Compare the numbers.
Now we have:
- \(2^{50}\)
- \(3^{40}\)
- \(2^{60}\) (from \(4^{30}\))
- \(5^{20}\)

### Step 3: Convert \(3^{40}\) and \(5^{20}\) to base 2.
To compare \(3^{40}\) and \(5^{20}\) with \(2^{50}\) and \(2^{60}\), we can use logarithms:

1. For \(3^{40}\):
\[ 3^{40} = 2^{40 \log_2(3)} \]
2. For \(5^{20}\):
\[ 5^{20} = 2^{20 \log_2(5)} \]

### Step 4: Calculate the logarithmic values.
Using approximate logarithmic values:
- \(\log_2(3) \approx 1.585\)
- \(\log_2(5) \approx 2.321\)

Now we can calculate:
1. \(40 \log_2(3) \approx 40 \times 1.585 = 63.4\)
2. \(20 \log_2(5) \approx 20 \times 2.321 = 46.42\)

### Step 5: Compare all the values.
Now we compare:
- \(2^{50}\) (which is \(50\))
- \(3^{40}\) (which is approximately \(63.4\))
- \(4^{30}\) (which is \(60\))
- \(5^{20}\) (which is approximately \(46.42\))

### Step 6: Determine the greatest number.
From the comparisons:
- \(3^{40} \approx 63.4\) is the greatest among \(50\), \(63.4\), \(60\), and \(46.42\).

### Conclusion:
Thus, the greatest number among \(2^{50}\), \(3^{40}\), \(4^{30}\), and \(5^{20}\) is \(3^{40}\).

### Final Answer:
The greatest number is \(3^{40}\).

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