`10^(11)` or `11^(10)` which is the greater number?

Options

A. Both are equal
B. `10^(11)gt11^(10)`
C. `11^(10)gt10^(11)`
D. Cannot be determined with out using tables

Explanation

To determine which is greater between \(10^{11}\) and \(11^{10}\), we can follow these steps:

### Step 1: Understand the expressions
We have two expressions:
- \(10^{11}\)
- \(11^{10}\)

### Step 2: Compare the bases and exponents
Here, the base of the first expression (10) is less than the base of the second expression (11), but the exponent of the first expression (11) is greater than the exponent of the second expression (10).

### Step 3: Apply the comparison rule
There is a mathematical rule that states if you have two numbers \(a\) and \(b\) (where \(b > a\)), and you compare \(a^b\) and \(b^a\), the expression with the larger exponent will generally be greater.

In our case:
- \(a = 10\)
- \(b = 11\)

Since \(b > a\), we can conclude that:
- \(10^{11}\) (the smaller base with the larger exponent) will be greater than \(11^{10}\) (the larger base with the smaller exponent).

### Step 4: Conclusion
Thus, we can conclude that:
\[ 10^{11} > 11^{10} \]

### Final Answer
The greater number is \(10^{11}\).

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