Let k = 10^(10). Then k is 1 followed by 10 zeros. The number k^k is 1 followed by how...

1. Translate the problem requirements: We need to understand that \(\mathrm{k = 10^{10}}\) means 1 followed by 10 zeros, and we need to find how many zeros follow the 1 in \(\mathrm{k^k = (10^{10})^{(10^{10})}}\)
2. Apply exponent multiplication rule: Use the power rule \(\mathrm{(a^m)^n = a^{(mn)}}\) to simplify \(\mathrm{k^k}\) into a single power of 10
3. Connect powers of 10 to zero counting: Recognize that \(\mathrm{10^n}\) always gives us 1 followed by exactly n zeros
4. Calculate the final exponent: Determine the value of the exponent to find the number of zeros

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're dealing with in everyday terms. We're told that \(\mathrm{k = 10^{10}}\), which means k is the number 1 followed by 10 zeros. So \(\mathrm{k = 10,000,000,000}\) (that's 10 billion).

Now we need to find \(\mathrm{k^k}\), which means we're raising k to the power of k itself. In other words, we're calculating \(\mathrm{(10^{10})^{(10^{10})}}\).

The question asks: how many zeros follow the 1 in this enormous number?

Process Skill: TRANSLATE - Converting the problem language into mathematical understanding

2. Apply exponent multiplication rule

Here's where we use a fundamental rule about exponents. When we have a power raised to another power, like \(\mathrm{(a^m)^n}\), we multiply the exponents together to get \(\mathrm{a^{(m×n)}}\).

So our expression \(\mathrm{(10^{10})^{(10^{10})}}\) becomes:
\(\mathrm{10^{(10 × 10^{10})}}\)

Now, what is \(\mathrm{10 × 10^{10}}\)? Well, \(\mathrm{10^{10}}\) means 1 followed by 10 zeros, so:
\(\mathrm{10 × 10^{10} = 10^1 × 10^{10} = 10^{(1+10)} = 10^{11}}\)

Therefore: \(\mathrm{k^k = (10^{10})^{(10^{10})} = 10^{(10^{11})}}\)

3. Connect powers of 10 to zero counting

Here's the key insight: any number written as \(\mathrm{10^n}\) always gives us exactly 1 followed by n zeros.

For example:

• \(\mathrm{10^1 = 10}\) (1 followed by 1 zero)
• \(\mathrm{10^2 = 100}\) (1 followed by 2 zeros)
• \(\mathrm{10^3 = 1,000}\) (1 followed by 3 zeros)

So if we have \(\mathrm{10^{(10^{11})}}\), this means we have 1 followed by exactly \(\mathrm{10^{11}}\) zeros.

4. Calculate the final exponent

Our final answer is that \(\mathrm{k^k = 10^{(10^{11})}}\), which means \(\mathrm{k^k}\) is 1 followed by \(\mathrm{10^{11}}\) zeros.

Looking at our answer choices:

• 20 - Way too small
• 100 - Still too small
• \(\mathrm{10^{10}}\) - This is just k itself, not nearly big enough
• \(\mathrm{10^{11}}\) - This matches our result!
• \(\mathrm{10^{20}}\) - This would be if we had \(\mathrm{(10^{10})^{20}}\), but that's not our problem

Final Answer

The number \(\mathrm{k^k}\) is 1 followed by \(\mathrm{10^{11}}\) zeros.

The correct answer is D: \(\mathrm{10^{11}}\).

Common Faltering Points

Errors while devising the approach

Students might confuse the relationship between powers of 10 and the number of zeros. They may think that \(\mathrm{10^n}\) has n+1 digits total instead of recognizing that \(\mathrm{10^n}\) is specifically "1 followed by n zeros." This fundamental misunderstanding would lead them to set up the entire problem incorrectly.

Some students might try to compute the actual numerical value of \(\mathrm{(10^{10})^{(10^{10})}}\) rather than working with exponent rules. Since this number is astronomically large, this approach is both impossible and unnecessary. The key insight is to work symbolically with powers of 10.

Errors while executing the approach

When simplifying \(\mathrm{(10^{10})^{(10^{10})}}\), students might incorrectly apply the exponent rule. Instead of multiplying the exponents to get \(\mathrm{10^{(10 × 10^{10})}}\), they might add them to get \(\mathrm{10^{(10 + 10^{10})}}\), which would lead to a completely different answer.

Students might make an error when computing \(\mathrm{10 × 10^{10}}\). Instead of recognizing this as \(\mathrm{10^1 × 10^{10} = 10^{11}}\), they might incorrectly calculate it as \(\mathrm{10^{100}}\) or some other value, leading to the wrong final exponent.

Errors while selecting the answer

After correctly determining that \(\mathrm{k^k = 10^{(10^{11})}}\), students might look at the answer choices and select \(\mathrm{10^{10}}\) (choice C) because it seems like a large number related to the original value of k. They fail to recognize that \(\mathrm{10^{11}}\) is vastly larger than \(\mathrm{10^{10}}\) and represents the correct number of zeros.