I need help

I need help with understanding the explanation behind a problem here

p=432234 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=49281=218281=263=18(210)618(103)6=10188. 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<1010 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
1 month ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

Sort by:

Top Newest

263=(23)(210)(26)2^{-63} = (2^{-3})(2^{10})(2^{-6}) since we add exponents when multiplying with like bases.

23=123=182^{-3} = \frac{1}{2^3} = \frac{1}{8}

Hope that helps.

David Stiff - 1 month ago

Log in to reply

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.

Brian Harahus - 4 weeks, 1 day ago

Log in to reply

Oops. Sorry, I think I made a small mistake. I should have said 263=(23)(260)2^{-63} = (2^{-3})(2^{-60}) (since we can add the exponents), and then we can further expand 2602^{-60} to get (23)(210)6(2^{-3})(2^{10})^{-6}. Does that make sense?

David Stiff - 4 weeks ago

Log in to reply

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

Brian Harahus - 3 weeks, 6 days ago

Log in to reply

@Brian Harahus Sorry about that. :) Glad it helped.

David Stiff - 3 weeks, 6 days ago

Log in to reply

Because 210=1024103    (210)6(103)62^{10}=1024\approx10^3\implies(2^{10})^{-6}\approx(10^3)^{-6}

Páll Márton (no activity) - 3 weeks, 3 days ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...