Finding the value of a square root

**dumluck** · May 7th 2011, 06:12 AM

What would be the quickest method to estimate (i.e to the nearest integer) the value of $3\sqrt{-89}$ ?

If 89 has a factor of 3 it wouldn't be too bad but given that its a prime this question seems a little difficult.

thx

**CaptainBlack** · May 7th 2011, 06:34 AM

Originally Posted by dumluck

What would be the quickest method to estimate (i.e to the nearest integer) the value of $3\sqrt{-89}$ ?

If 89 has a factor of 3 it wouldn't be too bad but given that its a prime this question seems a little difficult.

thx

$3 \sqrt{-89}=3i \sqrt{89}=-3i \sqrt{10^2-11}$

............. $=- 30 i \sqrt{1+11/100} \approx 30 i (1-(11/100)(1/2)+..)$

CB

**Wilmer** · May 7th 2011, 07:02 AM

For a really "rough" estimate:
sqrt(100) = 10
sqrt(81) = 9
10 > sqrt(89) > 9 ; get my drift?

**dumluck** · May 7th 2011, 07:10 AM

Originally Posted by Wilmer

For a really "rough" estimate:
sqrt(100) = 10
sqrt(81) = 9
10 > sqrt(89) > 9 ; get my drift?

Thanks Wilmer, that's the level I want to be at. However the answer is between -4 and -5?

**dumluck** · May 7th 2011, 07:22 AM

Ok I think I have it 4.4.4 = 64 5.5.5 = 125 therefore it is between -4 and -5.

**CaptainBlack** · May 7th 2011, 08:20 AM

Originally Posted by dumluck

Ok I think I have it 4.4.4 = 64 5.5.5 = 125 therefore it is between -4 and -5.

I'm sorry but it is not between any real numbers it is imaginary. Alternatively you have posted something other than what you think you have.

CB

**HallsofIvy** · May 7th 2011, 08:37 AM

Originally Posted by dumluck

Thanks Wilmer, that's the level I want to be at. However the answer is between -4 and -5?

The fact that you wrote " $3\sqrt{-89}$ " and titled this "Finding the value of a square root" made it very difficult for people to figure out that you really wanted the cube root, $\sqrt[3]{-89}$ .
$-4^3= -64$ and $-5^3= -125$ so the cube root of -89 is between -4 and -5.

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