If $k = 3^{81}$ and $k^k = 3^m$ , what is the value of $m$ ?

Answer Choices

Correct answer marked below

8^5

24^{381}

3^{54}

3^{85}

Correct

3^{72}

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Comments

ArjunMay 14

Wait, I'm confused. If k = 3^81 and k^k = 3^m, then setting the exponents equal gives k * log(k) = m? But the answer choices are all powers of 3, so maybe m is also a power of 3? I keep getting something like 3^81 * 81 * ln(3) / ln(3)? Not sure how to get a clean power of 3.

DivaMay 24

You're on the right track! Remember that k is itself a power of 3, so when you raise k to the k, you can write it as (3^81)^k. Then simplify the exponent.

KevinMay 19

Oh right, I forgot to substitute k = 3^81 directly into the exponent. So 3^{81k} = 3^m, so m = 81k = 81 * 3^81. But that's not matching the answer choices... hmm.

SakshiJun 1

This is a classic exponent manipulation. The key is to express k as a power of 3, then equate the exponents. But be careful with the exponent rules—it's easy to mess up multiplying vs adding exponents.

JohnMay 28

I got m = 3^85, but that seems too straightforward? Did anyone else get that? I feel like I might have made a careless mistake with the exponent addition: 81 + 81? No, wait, that would be 3^162. I'm second-guessing myself.

AleenaJun 6

Don't overthink it. If k = 3^81, then k^k = (3^81)^{3^81} = 3^{81 * 3^81}. So m = 81 * 3^81. Now write 81 as 3^4, so m = 3^4 * 3^81 = 3^{85}. That matches one of the choices.

DhruvAug 12

Thanks, this was helpful! I was stuck because I tried to take logs and got a mess. Substituting k = 3^81 directly makes it much simpler.

What this question tests

What it tests

Your grasp of exponent rules — multiplying and dividing powers, power-of-a-power, and negative/zero exponents.

Common trap

Adding exponents when multiplying bases that have the same exponent, or confusing (x^a)^b with x^(a·b).

If k=381k = 3^{81}k=381 and kk=3mk^k = 3^mkk=3m, what is the value of mmm?