Convert decimal into a square root number

[Redneck Aussie]

Hey lads.

I have a number 30.92329219. Instead of writing this long irrational number how do I convert this into an exact whole number square root? The square root of this number is 5.560871532.

If I times 30.92329219 by itself 4 times I get 914414.0622 which is close to a whole number. If I write the square root of 914414/172 I get 5.559592833. Its close but not the exact value. If I do the square root of 31 I get 5.567764363 which is not the same either.

I don't want the answer but can I get some advice please how to solve this? What am I missing here?

Cheers

 

[Steven G]

Ah finally an easy one. Just square 30.92329219 and put it under a square root sign. The problem you will have though is that 30.9232919² will not be a whole number.

Why would you thing that for every decimal number say between 1 and 2 you can express it as the sqrt(whole number).

sqrt(1) = 1 and sqrt(4)=2. There are only a few whole numbers between 1 and 4, namely 2 and 3. So sure some decimal numbers are real close to sqrt(2) and others are close to sqrt(3) but many many decimal numbers between 1 and 2 are not very close to sqrt(2) or sqrt(3).

More importantly NO ending decimal number equals sqrt(2) or sqrt(3). Sqrt(2) and sqrt(3) are irrational numbers which means they can not be expressed as terminating decimal numbers.

I hope this helps.

 

[topsquark]

Hey lads.

I have a number 30.92329219. Instead of writing this long irrational number how do I convert this into an exact whole number square root? The square root of this number is 5.560871532.

If I times 30.92329219 by itself 4 times I get 914414.0622 which is close to a whole number. If I write the square root of 914414/172 I get 5.559592833. Its close but not the exact value. If I do the square root of 31 I get 5.567764363 which is not the same either.

I don't want the answer but can I get some advice please how to solve this? What am I missing here?

Cheers

Given a random number you can't always write it in terms of a radical or a power. Sometimes you are just stuck with the decimals.

Is there some reason for you to think that your number should do so?

-Dan

 

[Deleted member 4993]

Hey lads.

I have a number 30.92329219. Instead of writing this long irrational number how do I convert this into an exact whole number square root? The square root of this number is 5.560871532.

If I times 30.92329219 by itself 4 times I get 914414.0622 which is close to a whole number. If I write the square root of 914414/172 I get 5.559592833. Its close but not the exact value. If I do the square root of 31 I get 5.567764363 which is not the same either.

I don't want the answer but can I get some advice please how to solve this? What am I missing here?

Cheers

30.92329219 = 3092329219E-08

and continue .....

 

[Cubist]

Ah finally an easy one. Just square 30.92329219 and put it under a square root sign. The problem you will have though is that 30.9232919² will not be a whole number.

Let x=30.92329219

Using Jomo's advice, it turns out that x² is actually very close to (y+1/4) where y is a natural number. Therefore, with a little thought x can be written as

[math] \frac{a\sqrt{b}}{2} [/math] where a and b are natural numbers, can you find them?

Spoiler: strong hint

[math] x ≈ \sqrt{y+1/4} = \frac{ \sqrt{4 \times(y+1/4)} }{\sqrt{4}} = \frac{\sqrt{4y+1}}{2}[/math]Then you'll find that 4y+1 has repeated factors that can be taken outside the sqrt

NOTE: The apparent accuracy of this approximation could be coincidental. This does not necessarily mean that the decimal x that you posted was obtained via this surd. (Although it seems massively likely!)

 

[Dr.Peterson]

I have a number 30.92329219. Instead of writing this long irrational number how do I convert this into an exact whole number square root? The square root of this number is 5.560871532.

The big question is, where did this number come from?

I wonder if you obtained it by solving some problem using decimals, and found that the "correct" answer was a radical expression, and you think that there should be a way to obtain the latter from the former.

As has been said, there is no general method for that, because most irrational numbers can't be exactly expressed as radicals.

But if you were expected to get an exact answer, the right thing to do is to go back through your work, simplifying radicals exactly rather than using decimal approximations, and you should get the exact answer.

Is my guess close to being correct?

 

[Harry_the_cat]

Sadly, I've been left out again.

 

[Redneck Aussie]

Ok so what you all are saying is not every number can be converted from an irrational number in to a radical?

That number 30.92329219 is the base of my triangle which represents a dam wall in meters. The hypotenuse is 34.64101615. But I can write that as 10 square 12 and I get the same answer. I just thought I could convert it into a more pleasant looking figure.

 

[Cubist]

I just thought I could convert it into a more pleasant looking figure.

You can, maybe you missed my post #5? You said in post #1 "I don't want the answer" however if you're actually after an answer then here it is:-

Spoiler: answer

[math] \frac{15\sqrt{17}}{2} ≈ 30.923292192[/math]

That number 30.92329219 is the base of my triangle which represents a dam wall in meters. The hypotenuse is 34.64101615.

But where did these decimal figures come from? Are they actually written like that in a homework assignment that you were given, or did you produce them yourself using a calculator? If you got them via a calculator then please post back with the complete problem because we can show you how to avoid this situation.

 

[HallsofIvy]

And, by the way, "30.92329219" is NOT an "irrational number"!

 

[Dr.Peterson]

Ok so what you all are saying is not every number can be converted from an irrational number in to a radical?

That number 30.92329219 is the base of my triangle which represents a dam wall in meters. The hypotenuse is 34.64101615. But I can write that as 10 square 12 and I get the same answer. I just thought I could convert it into a more pleasant looking figure.

You should be aware that [MATH]10\sqrt{12}[/MATH] (which is called 10 square root 12, not 10 square 12) is equal to 34.641016151377545870548926830117..., so your 34.64101615 is only an approximation. Moreover, no measured value could be that precise, much less be expressed exactly by a radical. Clearly this is from a classroom problem, not from reality (unless perhaps your numbers are both derived from some whole-number measurements, in which case the problem is that significant digits have been ignored).

If your numbers are taken as approximations of the exact numbers [MATH]\frac{15\sqrt{17}}{2}[/MATH] and [MATH]10\sqrt{12}[/MATH], then the other leg can be calculated exactly as [MATH]\sqrt{\left(10\sqrt{12}\right)^2 - \left(\frac{15\sqrt{17}}{2}\right)^2} = \sqrt{1200 - \frac{225\cdot17}{4}} = \dots[/MATH]

 

[Steven G]

And, by the way, "30.92329219" is NOT an "irrational number"!

To OP: Radical like sqrt(3), sqrt(5), sqrt(6), sqrt (101), sqrt(211), ... are all irrational.

 

[Redneck Aussie]

Oh, Wow cubist that's it.


So how do I make 30.92329219 into 15 square root 17 divide 2?

I am still a little confused how you did that?

I turned 34.64101615 into 10 square root 12 by multiplication and then factoring.


Thank you for your help everybody. I appreciate it.

 

[Cubist]

I am still a little confused how you did that?

Following the method in post#5 (which was inspired by @Jomo)

30.92329219² ≈ 956¼

Then we can take the square root of both sides

[math] 30.92329219 ≈ \sqrt{956\tfrac{1}{4}} [/math]
Then eliminate the quarter [math] \sqrt{956\tfrac{1}{4}} = \frac{ \sqrt{4 \times(956\frac{1}{4})} }{\sqrt{4}} = \frac{\sqrt{3825}}{2} [/math]
Take repeated factors outside [math] \frac{\sqrt{3825}}{2} = \frac{\sqrt{3 \times 3 \times 5 \times 5 \times 17}}{2} = \frac{3 \times 5 \times \sqrt{17}}{2} = \frac{15 \sqrt{17}}{2}[/math]
Sorry I don't have time to look into your triangle image at the moment, but perhaps someone else can help you with that (if not then I'll take a look later)

 

[HallsofIvy]

First, looking at your picture, that does NOT appear to be "10 square root of 12", \(\displaystyle 10\sqrt{12}\), it appears to be "the 10th root or 12", \(\displaystyle \sqrt[10]{12}\). I wondered about that from the start since \(\displaystyle 10\sqrt{12}\) would be better written \(\displaystyle 20\sqrt{3}\). (\(\displaystyle sqrt{12}= \sqrt{4(3)}= \sqrt{4}\sqrt{3}= 2\sqrt{3}\).) Using a calculator, the tenth root of 12 is approximately 1.282. And \(\displaystyle 10\sqrt{12}= 20\sqrt{3}\) is approximately 34.641, NOT 30.92329219!

Looking at the picture you finally posted, you have a right triangle with one angle 30 degrees, hypotenuse of length \(\displaystyle \sqrt[10]{12}\) and apparently are trying to find the "near side". The cosine or an angle is defined as "near side over hypotenuse", so \(\displaystyle \frac{x}{\sqrt[10]{12}}= cos(30)\). The cosine of 30 degrees is \(\displaystyle \frac{\sqrt{3}}{2}\) so \(\displaystyle \frac{x}{\sqrt[10]{12}}= \frac{\sqrt{3}}{2}\) so \(\displaystyle x= \frac{\sqrt{3}\sqrt[10]{12}}{2}\).

If the length of the hypotenuse really is \(\displaystyle 10\sqrt{12}= 20\sqrt{3}\) then the length of the "near side" is \(\displaystyle \left(20\sqrt{3}\right)\left(\sqrt{3}{2}\right )= (10)(3)= 30\) which at least is closer to your "30.92329219".

I suggest you talk to your teacher about this problem. You seem to be completely confused about it. Certainly I have no idea what you are trying to find!

 

[Dr.Peterson]

I turned 34.64101615 into 10 square root 12 by multiplication and then factoring.

Thank you for your help everybody. I appreciate it.

Please state the entire actual problem as given to you. The picture doesn't clarify where that excessively precise number on the bottom came from, or why (assuming HallsofIvy is right) the hypotenuse would be given as a tenth root. We really need to see the whole thing (and that's what we ask for: see here).