# Thread: Square Root = Two Answers

1. ## Square Root = Two Answers

Taking the square root always yields a positive and negative answer. Why is this the case?

Example:

root {9} = -3 and 3.

2. ## Re: Square Root = Two Answers

Originally Posted by mathdad1965
Taking the square root always yields a positive and negative answer. Why is this the case?

Example:

root {9} = -3 and 3.
this is incorrect.

The square root function always returns a positive value.

While it's true $3^2 = 9$ and that $(-3)^2 = 9$

$\sqrt{9} = 3,~\sqrt{9} \neq -3$

3. ## Re: Square Root = Two Answers

The square root function only yields one answer $\displaystyle \sqrt{x^2}=|x|$.

But there are two numbers that have a square equal to $\displaystyle x^2$. Namely $\displaystyle x$ and $\displaystyle -x$. So the reverse operation to squaring has two possible results.

In your particular case $\displaystyle 3^2=(-3)^2=9$.

4. ## Re: Square Root = Two Answers

Originally Posted by romsek
this is incorrect.

The square root function always returns a positive value.
positive should of course be "non-negative"

$\sqrt{0} = 0$

5. ## Re: Square Root = Two Answers

Thanks. I always thought that the square root of 9 = -3 and 3.

6. ## Re: Square Root = Two Answers

Both -3 and 3 are square roots of 9. The operator/function $\sqrt{.}$ is defined to return the non-negative square root of its argument (or the principle value if complex)

7. ## Re: Square Root = Two Answers

Originally Posted by zzephod
Both -3 and 3 are square roots of 9. The operator/function $\sqrt{.}$ is defined to return the non-negative square root of its argument (or the principle value if complex)
Thanks.

8. ## Re: Square Root = Two Answers

Notice the difference in wording. "Both -3 and 3 are square roots of 9." "The square root of 9 is 3."

9. ## Re: Square Root = Two Answers

Originally Posted by HallsofIvy
Notice the difference in wording. "Both -3 and 3 are square roots of 9." "The square root of 9 is 3."
I see that -3 is not a square root of 9. We cannot take the square root of negative numbers.

10. ## Re: Square Root = Two Answers

Originally Posted by zzephod
Both -3 and 3 are square roots of 9. The operator/function $\sqrt{.}$ is defined to return the non-negative square root of its argument (or the principle value if complex)
out of curiosity what do you think are the value(s) of $9^{1/2}$

i.e. the concept of the square root without the $\sqrt{\text{ }}$ operator

11. ## Re: Square Root = Two Answers

Originally Posted by mathdad1965
I see that -3 is not a square root of 9. We cannot take the square root of negative numbers.
Then you are not understanding. You are taking the square root of 9, a positive number. You are NOT taking the square root of -3.

12. ## Re: Square Root = Two Answers

Originally Posted by HallsofIvy
Then you are not understanding. You are taking the square root of 9, a positive number. You are NOT taking the square root of -3.
I get it. The sqrt {positive number} yields a positive answer.

root {4} = 2

root {16} = 4

root {25} = 5 and so on.........

13. ## Re: Square Root = Two Answers

Originally Posted by romsek
out of curiosity what do you think are the value(s) of $9^{1/2}$

i.e. the concept of the square root without the $\sqrt{\text{ }}$ operator
It depends on the context. Without qualification I would take it to denote the (two) distinct values of $3e^{n\pi i};\ n\in \mathbb{Z}$, because that is the interpretation in the contexts that I normally work with.

.

14. ## Re: Square Root = Two Answers

Originally Posted by zzephod
It depends on the context. Without qualification I would take it to denote the (two) distinct values of $3e^{n\pi i};\ n\in \mathbb{Z}$, because that is the interpretation in the contexts that I normally work with.

.
There is no context to consider. You are being asked straight-up for value(s) of $\displaystyle \ \ 9^{1/2}.$

Look at these steps worked out here at:

www.quickmath.com

Type: 9^(1/2)

Then click "Simplify" and look at the steps.

15. ## Re: Square Root = Two Answers

Originally Posted by greg1313
There is no context to consider. You are being asked straight-up for value(s) of $\displaystyle \ \ 9^{1/2}.$

Look at these steps worked out here at:

www.quickmath.com

Type: 9^(1/2)

Then click "Simplify" and look at the steps.
that doesn't tell the whole story. In the second step you can replace

$(3^2)^{1/2}$

with

$((-3)^2)^{1/2}$

and end up with $-3$ as an answer

However Mathematica says that the only answer is $9^{1/2}=3$ even when expressed as $\left(9 e^{i 2 \pi}\right)^{1/2}$

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