The greatest number among `3^50, 4^40, 5^30`, and `6^20` is

Options

A. `4^40`
B. `5^30`
C. `6^20`
D. `3^50`

Answer

To find the greatest number among \(3^{50}\), \(4^{40}\), \(5^{30}\), and \(6^{20}\), we can compare these numbers by expressing them in a common base or by simplifying their powers.

### Step 1: Normalize the powers
We can normalize the powers by dividing each exponent by the smallest exponent among them, which is \(20\) (the exponent of \(6^{20}\)).

- For \(3^{50}\):
\[ 3^{50} = (3^{50/20})^{20} = (3^{2.5})^{20} \]

- For \(4^{40}\):
\[ 4^{40} = (4^{40/20})^{20} = (4^{2})^{20} \]

- For \(5^{30}\):
\[ 5^{30} = (5^{30/20})^{20} = (5^{1.5})^{20} \]

- For \(6^{20}\):
\[ 6^{20} = (6^{20/20})^{20} = (6^{1})^{20} \]

### Step 2: Calculate the new bases
Now we calculate the new bases:

- \(3^{2.5} = 3^{5/2} = \sqrt{3^5} = \sqrt{243} \approx 15.588\)
- \(4^{2} = 16\)
- \(5^{1.5} = 5^{3/2} = \sqrt{5^3} = \sqrt{125} \approx 11.180\)
- \(6^{1} = 6\)

### Step 3: Compare the values
Now we compare the values of \(3^{2.5}\), \(4^{2}\), \(5^{1.5}\), and \(6^{1}\):

- \(3^{2.5} \approx 15.588\)
- \(4^{2} = 16\)
- \(5^{1.5} \approx 11.180\)
- \(6^{1} = 6\)

### Step 4: Identify the greatest value
From the calculated values, we can see that \(4^{2} = 16\) is the greatest among the four.

### Conclusion
Thus, the greatest number among \(3^{50}\), \(4^{40}\), \(5^{30}\), and \(6^{20}\) is:
\[ \boxed{4^{40}} \]