# I need help

I need help with understanding the explanation behind a problem here

$p=\frac { { 4 }^{ 3^{ 2 } } }{ { 2 }^{ 3^{ 4 } } }$

Which of the following is correct about the above number P?

Here's the explanation Brilliant gave me.

$P\quad =\quad \frac { { 4 }^{ 9 } }{ { 2 }^{ 81 } } \quad =\quad \frac { { 2 }^{ 18 } }{ { 2 }^{ 81 } } \quad =\quad { 2 }^{ -63 }\quad =\quad \frac { 1 }{ 8 } ({ 2 }^{ 10 })^{ -6 }\quad \approx \quad \frac { 1 }{ 8 } (10^{ 3 })^{ -6 }\quad =\quad \quad \frac { { 10 }^{ -18 } }{ 8 } .\quad$

The answer: $P\quad <\quad { 10 }^{ -10 }\quad$

I particularly want help with how you would go about getting the 1/8* (2^10)^-6 in the simplified expression. All I really need is some clarification in the explanation. Thanks ahead of time. Note by Brian Harahus
2 months ago

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Because $2^{10}=1024\approx10^3\implies(2^{10})^{-6}\approx(10^3)^{-6}$

$2^{-63} = (2^{-3})(2^{10})(2^{-6})$ since we add exponents when multiplying with like bases.

$2^{-3} = \frac{1}{2^3} = \frac{1}{8}$

Hope that helps.

- 2 months ago

That brings up another question I have, How did you get 2^-3 * 2^10 * 2^-6? I'm lost on how you got that.

- 2 months ago

Oops. Sorry, I think I made a small mistake. I should have said $2^{-63} = (2^{-3})(2^{-60})$ (since we can add the exponents), and then we can further expand $2^{-60}$ to get $(2^{-3})(2^{10})^{-6}$. Does that make sense?

- 2 months ago

Yes, thank you, that was a huge help. I was getting kind of confused there for a minute

- 2 months ago