Confusing answer from Wolfram Alpha

[Kruidnootje]

\sqrt{\sqrt{}}Firstly, I have tried to work out the square root sign, I would have thought this common sign to be listed on the right.

While checking my answer to the square root of 171 I noticed wolfram displayed also another function under the listing of "exact answer". It gave 3 with a square root sign then 19. A search on the web yielded three different explanations. One I did not understand the other two as follows:

a) 3* the square root - to be expected
b)The inverse function of aⁿ - something new I just learned

Needless to say neither of the above appear correct, or wolfram means something completely different.

 

[topsquark]

\sqrt{\sqrt{}}Firstly, I have tried to work out the square root sign, I would have thought this common sign to be listed on the right.

While checking my answer to the square root of 171 I noticed wolfram displayed also another function under the listing of "exact answer". It gave 3 with a square root sign then 19. A search on the web yielded three different explanations. One I did not understand the other two as follows:

a) 3* the square root - to be expected
b)The inverse function of aⁿ - something new I just learned

Needless to say neither of the above appear correct, or wolfram means something completely different.

Yes, the LaTeX code would be \sqrt{ }. [math]\sqrt{2}[/math] gives \(\displaystyle \sqrt{2}\).

\(\displaystyle \sqrt{171} = \sqrt{9 \cdot 19} = \sqrt{9} \cdot \sqrt{19} = 3 \sqrt{19}\)

-Dan

 

[MarkFL]

\sqrt{\sqrt{}}Firstly, I have tried to work out the square root sign, I would have thought this common sign to be listed on the right.

It is:

View attachment 8046

 

[Kruidnootje]

It is:

2\sqrt{2}

Hallo, thankyou, I know I'm a pain here but I did see these and nothing comes up in preview mode except the code. As you can see rom my post I tried this and left the code in the message. Also my preview does not work so I keep posting and then re-editing to check that all the code is right.

 

[Kruidnootje]

Yes, the LaTeX code would be \sqrt{ }. [math]\sqrt{2}[/math] gives \(\displaystyle \sqrt{2}\).

\sqrt{171} = \sqrt{9 \cdot 19} = \sqrt{9} \cdot \sqrt{19} = 3 \sqrt{19}[/math]

-Dan

\sqrt.{}\(\displaystyle \sqrt(1000)\) Took all of breakfast and numerous edits but got there eventually, thankyou.

\(\displaystyle \sqrt(171)\) = 13.1 I recognised the 3*19 before I posted. \(\displaystyle \sqrt(9)\) * \(\displaystyle \sqrt(19)\) = 13.1 Ok Logical so far...But that 3 \(\displaystyle \sqrt(19)\) makes no sense still?. Sorry, I have racked my brains trying to get a simple 13 out of that ! It's not a cube root, not a multiplication by 3...

 

[I like Serena]

2\sqrt{2}

Hallo, thankyou, I know I'm a pain here but I did see these and nothing comes up in preview mode except the code. As you can see rom my post I tried this and left the code in the message. Also my preview does not work so I keep posting and then re-editing to check that all the code is right.

It's $\LaTeX$, which means that we need for instance \$ symbols around it, which is all there is to it.
If you type \$2\sqrt{2}\$ in the $\LaTeX$ Live Preview, you'll see that it works.
And if you hover over the $\tiny\boxed\sum$ quick button, you'll see that you can use it to turn text into $\LaTeX$.

 

[Kruidnootje]

It's $\LaTeX$, which means that we need for instance \$ symbols around it, which is all there is to it.
If you type \$2\sqrt{2}\$ in the $\LaTeX$ Live Preview, you'll see that it works.
And if you hover over the $\tiny\boxed\sum$ quick button, you'll see that you can use it to turn text into $\LaTeX$.

Thankyou for your help. I'll make a list of common codings in a text file and then copy and paste whenever I need them.

 

[topsquark]

\sqrt.{}\(\displaystyle \sqrt(1000)\) Took all of breakfast and numerous edits but got there eventually, thankyou.

\(\displaystyle \sqrt(171)\) = 13.1 I recognised the 3*19 before I posted. \(\displaystyle \sqrt(9)\) * \(\displaystyle \sqrt(19)\) = 13.1 Ok Logical so far...But that 3 \(\displaystyle \sqrt(19)\) makes no sense still?. Sorry, I have racked my brains trying to get a simple 13 out of that ! It's not a cube root, not a multiplication by 3...

I'm not sure what you are asking for. \(\displaystyle 3 \sqrt{19} \approx 3 \cdot 4.395 \approx 13.077\). That's just calculator work.

-Dan

 

[Kruidnootje]

I'm not sure what you are asking for. \(\displaystyle 3 \sqrt{19} \approx 3 \cdot 4.395 \approx 13.077\). That's just calculator work.

-Dan

\(\displaystyle 3 \sqrt{19}\) essentially means: (3*4.359) x (3*4.359) = 171

Now I understand it! So whenever you see a number preceding the SQRT sign you simply SQRT the number that is asking to be SQRT'ed IE \(\displaystyle \sqrt{19}\) then multiply that by the number preceding the SQRT sign then multiply the resultant two together. That was NEVER intuitive to work out. Wolfram Alpha threw up \(\displaystyle 3 \sqrt{19}\) = 171. That was the problem. Then on further research sites were saying that this meant the inverse of the root to the whatever power and so on...really making no sense.

 

[topsquark]

\(\displaystyle 3 \sqrt{19}\) essentially means: (3*4.359) x (3*4.359) = 171

Now I understand it! So whenever you see a number preceding the SQRT sign you simply SQRT the number that is asking to be SQRT'ed IE \(\displaystyle \sqrt{19}\) then multiply that by the number preceding the SQRT sign then multiply the resultant two together. That was NEVER intuitive to work out. Wolfram Alpha threw up \(\displaystyle 3 \sqrt{19}\) = 171. That was the problem. Then on further research sites were saying that this meant the inverse of the root to the whatever power and so on...really making no sense.

I'm glad you understand this but note that your description of how to get the answer is a bit off for many cases. For example, one way to approach this kind of problem is to find the prime factors of the number under the square root:
\(\displaystyle \sqrt{5175} = \sqrt{5^2 \cdot 3^2 \cdot 23} = \sqrt{ 5^2 } \cdot \sqrt{ 3^2 } \cdot \sqrt{23} = 5 \cdot 3 \cdot \sqrt{23} = 15 \sqrt{23} \)

-Dan

 

[Kruidnootje]

I'm glad you understand this but note that your description of how to get the answer is a bit off for many cases. For example, one way to approach this kind of problem is to find the prime factors of the number under the square root:
\(\displaystyle \sqrt{5175} = \sqrt{5^2 \cdot 3^2 \cdot 23} = \sqrt{ 5^2 } \cdot \sqrt{ 3^2 } \cdot \sqrt{23} = 5 \cdot 3 \cdot \sqrt{23} = 15 \sqrt{23} \)

-Dan

That's all fine Dan, but not possible for someone like me for I have no idea how to do that with large numbers. Unless you care to share with me a formula for finding these factors? Looking up in wolfram I see that the only factors for 171 are 3 19 and 57. So I presume all that you have is: 3³ *19 and that's the answer?

 

[I like Serena]

That's all fine Dan, but not possible for someone like me for I have no idea how to do that with large numbers. Unless you care to share with me a formula for finding these factors? Looking up in wolfram I see that the only factors for 171 are 3 19 and 57. So I presume all that you have is: 3³ *19 and that's the answer?

There are some tricks to it.

Is the number even? Then 2 is a factor. Divide by 2 and see what else pops out.

Does the number end in 0 or 5? Then 5 is a factor. Divide by 5 and keep going.

Is the sum of the digits divisible by 3? Then 3 is a factor.
Note that the sum of the digits of 171 is 1+7+1=9, implying that 3 (and also 9) are a factor.
Divide 171 by 9 and we're left with 19. So $171=3^2\cdot 19$.

 

[Kruidnootje]

There are some tricks to it.

Is the number even? Then 2 is a factor. Divide by 2 and see what else pops out.

Does the number end in 0 or 5? Then 5 is a factor. Divide by 5 and keep going.

Is the sum of the digits divisible by 3? Then 3 is a factor.
Note that the sum of the digits of 171 is 1+7+1=9, implying that 3 (and also 9) are a factor.
Divid 171 by 9 and we're left with 19. So $171=3^2\cdot 19$.

Tankyou, that's a good start, some practising on hand and I'll stick it to memory. However I was musing over this whole issue with the sqrt of 171 and wondering why on earth. Oh why why why did anyone ever come up with that manner of expressing the sqrt 171 as ³\(\displaystyle \sqrt{19}\) just seems a useless roundabout way of saying the simple 13.076

 

[I like Serena]

Tankyou, that's a good start, some practising on hand and I'll stick it to memory. However I was musing over this whole issue with the sqrt of 171 and wondering why on earth. Oh why why why did anyone ever come up with that manner of expressing the sqrt 171 as \(\displaystyle 3\sqrt{19}\)

It's called simplifying. Sometimes it helps to keep things simple.
Usually it's not required though. Unless specifically requested we can leave it as $\sqrt{171}$.
Making it 13.076 is a bit different though, since that's an approximation and no longer exact.

 

[Kruidnootje]

It's called simplifying. Sometimes it helps to keep things simple.
Usually it's not required though. Unless specifically requested we can leave it as $\sqrt{171}$.
Making it 13.076 is a bit different though, since that's an approximation and no longer exact.

well it took me 10 seconds to find the sqrt the old fashioned way, and 3 days to understand the simplified way...!

 

[I like Serena]

well it took me 10 seconds to find the sqrt the old fashioned way, and 3 days to understand the simplified way...!

I take it that you mean 'using a calculator' when you say the 'old fashioned way'?
Just saying, understanding the simplified way will be valuable later on. That's when we're no longer talking about mere numbers, but start introducing letters (variables).
It's not a wasted effort, and once you understand it, it may even be faster than using a calculator. And there's also a bit of fun involved. (Nerd)

 

[MarkFL]

well it took me 10 seconds to find the sqrt the old fashioned way, and 3 days to understand the simplified way...!

With calculators/computers the need to simplify isn't as great for finding a decimal approximation. If you are using manual algorithms to compute square roots, then such simplifications are very useful. :)

 

[Kruidnootje]

With calculators/computers the need to simplify isn't as great for finding a decimal approximation. If you are using manual algorithms to compute square roots, then such simplifications are very useful. :)

Yep and maybe one day I'll look back on this thread and be embarrassed by my moans.

 

[HOI]

\(\displaystyle 3 \sqrt{19}\) essentially means: (3*4.359) x (3*4.359) = 171

NO, it doesn't! \(\displaystyle (3 \sqrt{19})^2= 171\). Approximating \(\displaystyle \sqrt{19}\) by 4.359, 3(4.359) is approximately 13.077 and multiplying that by itself (squaring it) gives 171.007929, approximately 171. The exact value of \(\displaystyle \sqrt{171}\) is \(\displaystyle 3\sqrt{19}\).

Now I understand it! So whenever you see a number preceding the SQRT sign you simply SQRT the number that is asking to be SQRT'ed IE \(\displaystyle \sqrt{19}\) then multiply that by the number preceding the SQRT sign then multiply the resultant two together. That was NEVER intuitive to work out. Wolfram Alpha threw up \(\displaystyle 3 \sqrt{19}\) = 171. That was the problem. Then on further research sites were saying that this meant the inverse of the root to the whatever power and so on...really making no sense.

Why is it not "intuitive" that "ab" in algebra means "a times b"? You should have learned that when you were 11 or 12 years old. And "square root" is defined as "the inverse function to squaring". More generally, \(\displaystyle \sqrt[n]{x}\) is the inverse function to \(\displaystyle x^n\). What did you think "square root" meant?