Difficult Power Rule task (without calculator)

**kevin3000** · July 20th 2010, 08:57 AM

I really need help with this one.

**undefined** · July 20th 2010, 10:18 AM

2^7 = 128. So...

**1005** · July 20th 2010, 12:26 PM

Originally Posted by undefined

2^7 = 128. So...

How does that help him find an estimation for 128^48 without using a computer?
$128^{48}=(2^{14})^{24}=16384^{24}$

Or am I missing something? All I know is I wouldn't want to do 16384^14 by hand.

**undefined** · July 20th 2010, 12:29 PM

Originally Posted by 1005

How does that help him find an estimation for 128^48 without using a computer?
$128^{48}=(2^{14})^{24}=16384^{24}$

Or am I missing something? All I know is I wouldn't want to do 16384^14 by hand.

All we need is an estimate. Why not take

$128^{48}=16384^{24} \approx (10^4)^{24} = 10^{96}$

?

**Soroban** · July 20th 2010, 02:20 PM

Hello, kevin3000!

This is a strange problem . . .

Calculate an approximate value to $128^{48}$
if you know that: $2^{14} = 16,\!384$ . . I don't see how this is useful.

I would do it like this . . .

We have: . $N \;=\;128^{48} \:=\:\left(2^7\right)^{48} \:=\:2^{336}$

Since $2^{10} \:=\:1024 \:\approx\:1000 \:=\:10^3$

we have: . $N \;=\;2^6\cdot2^{330} \;=\;2^6\cdot (2^{10})^{33}$

. . . . . . . . $N\;\approx\;2^6\cdot(10^3)^{33} \;=\;64\cdot10^{99}$

Therefore: . $N \;\approx\;6.4 \times 10^{100}$

**1005** · July 20th 2010, 02:45 PM

undefined's approximation(10^96) is 5 order of magnitude off with 99.99% error (lol)

and Soroban's approximation is 1 order of magnitude off with 54.28% error.

**undefined** · July 20th 2010, 03:11 PM

Originally Posted by 1005

undefined's approximation(10^96) is 5 order of magnitude off with 99.99% error (lol)

and Soroban's approximation is 1 order of magnitude off with 54.28% error.

It's funny yes but I was merely following the instructions of the problem (or trying to do so, since the problem doesn't make much sense); given freedom, I would also use the approximation 2^10 $\approx$ 10^3.

Anyway if you go high enough, 2^10 $\approx$ 10^3 will give you 99.99% error as well.

**bjhopper** · July 20th 2010, 05:37 PM

[QUOTE=1005;539205]undefined's approximation(10^96) is 5 order of magnitude off with 99.99% error (lol)

and Soroban's approximation is 1 order of magnitude off with 54.28% error.[/QUOTE

Soroban's approx is not one magnitude off. It can be expressed as 0.64E101. Calculator answer is 1.4xE 101

bjh

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