# If x^2 = root (y), then how to calculate the value of x?

#### Indranil

##### Junior Member
If x^2 = root (y), then how to calculate the value of 'x'?

#### tkhunny

##### Moderator
Staff member
If x^2 = root (y), then how to calculate the value of 'x'?
If it were x^2 = 4, what would you do?

#### mmm4444bot

##### Super Moderator
Staff member
One method is to take the square root of each side.

|x| = sqrt(√y)

x = ±sqrt(√y)

I'll let you think about what happens, when we take the square root of a square root. If you get stuck, try switching to exponential form. :cool:

#### mmm4444bot

##### Super Moderator
Staff member
If x^2 = roo͏t (y), then how to calculate the value of 'x'?
Be careful, using that notation for square root; some people might not understand.

sqrt(y) is standard notation.

Also, "calculate the value of x" is not what you're thinking. We cannot find a value for x, unless we're first given a value for y.

What you're asking is, "How to solve this equation for x?" :cool:

Last edited:

#### Indranil

##### Junior Member
If it were x^2 = 4, what would you do?
I would do it in two ways
1. x^2 = 4 = 2^2 = 2.
2. x^2 = 4 = √4 = 2

#### Indranil

##### Junior Member
One method is to take the square root of each side.

|x| = sqrt(√y)

x = ±sqrt(√y)

I'll let you think about what happens, when we take the square root of a square root. If you get stuck, try switching to exponential form. :cool:
x^2 = √y = y^(1/2)
so, x^2 = y^(1/2) Now What can I do?

#### Dr.Peterson

##### Elite Member
I would do it in two ways
1. x^2 = 4 = 2^2 = 2.
2. x^2 = 4 = √4 = 2
You are misusing the "=" sign so badly that what you write is meaningless. (This is, unfortunately, common among students who never learned what the equal sign really means.) It is not true that 2^2 = 2, or that 4 = √4. Please be careful to write what you mean.

I think what you mean is something like this:

1. x^2 = 4 = 2^2, so x = 2.
2. x^2 = 4, so x = √4 = 2

But it has already been pointed out that this does not give you the complete solution set, because you are ignoring the possibility that x could be negative. You can't blindly "take the square root", because there are two square roots.

The first way, you are imagining that, since x^2 = 2^2, the x must be 2 -- that is, assuming that the square function is one-to-one, so that any result can be attained in only one way. That is not true. There are two numbers whose square is 4, namely 2 and -2.

The second way, the square root of the left side, √(x^2), is not x, but |x|, because for example if x=-2, then √((-2)^2) = √4 = 2 = |-2|, not -2 itself.

#### Jomo

##### Elite Member
I would do it in two ways
1. x^2 = 4 = 2^2 = 2.
2. x^2 = 4 = √4 = 2
No, this is not true and I bet you know it! Equal signs MUST be valid.

1. x^2 = 4 = 2^2 = 2. NO, 2^2 does NOT equal 2!!!
2. x^2 = 4 = √4 = 2. NO, 4 does not equal √4 !!!

#### tkhunny

##### Moderator
Staff member
I would do it in two ways
1. x^2 = 4 = 2^2 = 2.
2. x^2 = 4 = √4 = 2
This I why we ask. Thank you.

First, don't do that. Please factor and solve completely so that you don't have to remember anything important. Let the notation help you.

x^2 = 4

x^2 - 4 = 0

(x-2)(x+2) = 0

x= 2 or x = -2

See how we managed BOTH solutions without any pain?

Okay, now why is your original problem ANY different? (Hint: It isn't.)

Last edited:

#### Indranil

##### Junior Member
One method is to take the square root of each side.

|x| = sqrt(√y)

x = ±sqrt(√y)

I'll let you think about what happens, when we take the square root of a square root. If you get stuck, try switching to exponential form. :cool:
x^2 = √y, x^2 = y^(1/2)
so, x^2 = y^(1/2) Now What can I do? Still, I am stuck in this problem. In this case, can how to calculate the value of 'x'?

Last edited:

#### Subhotosh Khan

##### Super Moderator
Staff member
x^2 = √y, x^2 = y^(1/2)
so, x^2 = y^(1/2) Now What can I do? Still, I am stuck in this problem. In this case, can how to calculate the value of 'x'?
If NOTHING ELSE is given to you - you have nothing to do of any consequence.

What else is given to you - like value of "y"?

Why aren't you posting the COMPLETE problem - as it was given to you?

#### Indranil

##### Junior Member
If NOTHING ELSE is given to you - you have nothing to do of any consequence.

What else is given to you - like value of "y"?

Why aren't you posting the COMPLETE problem - as it was given to you?
If x^2 = √16, then how to calculate the value of 'x' in exponential form? I have done below in the root form
x^2 = √16, x= √(√16), x= √4 =2 Is the method is correct? if correct, please solve this problem in exponential form. I have tried below
x^2 = 16^1/2, x^2 = 4, x^2 = 2^2, x = 2 Am I correct? If correct, can you please solve it in any other methods?

#### Dr.Peterson

##### Elite Member
If x^2 = √16, then how to calculate the value of 'x' in exponential form? I have done below in the root form
x^2 = √16, x= √(√16), x= √4 =2 Is the method is correct? if correct, please solve this problem in exponential form. I have tried below
x^2 = 16^1/2, x^2 = 4, x^2 = 2^2, x = 2 Am I correct? If correct, can you please solve it in any other methods?
First, you're not quite right, because x could be either +2 or -2, as has been empathized. PLEASE don't forget that; I think some of us may be thinking you are refusing to pay attention, and getting frustrated.

But for the moment, let's suppose you were also told that x>0, so you don't need to worry about that.

As written, x^2 = √16, I would first simplify the right side, so x^2 = 4. Then there are several ways to get to the answer; your way is valid only if you know that x>0! If you don't, then you need to either remember to include the negative case, or use the factoring method you have been shown:

x^2 = 4
x^2 - 4 = 0
(x + 2)(x - 2) = 0
x + 2 = 0 or x - 2 = 0
x = -2 or x = 2

But suppose now that you were given x^2 = √y, where you don't know y, and you are told to solve for x in terms of y, "in exponential form". Then you might do this, using exponents all the way:

x^2 = y^(1/2)
(x^2)^(1/2) = (y^(1/2))^(1/2)
x^(2*1/2) = y^(1/2 * 1/2)
x^1 = y^(1/4)
x = y^(1/4)

Now, in doing this, I assumed that x>0. If I couldn't do that, I would have to be aware that, just as there are two square roots, there are two 1/2 powers (since that means the same thing), so I would do this:

x^2 = y^(1/2)
(x^2)^(1/2) = ±(y^(1/2))^(1/2)
x^(2*1/2) = ±y^(1/2 * 1/2)
x^1 = ±y^(1/4)
x = ±y^(1/4)

(Note that the right side was given as √y, that means only the positive root, so I didn't use ± at the start.)

It would also be possible to express this in terms of factoring, though that is a little more awkward than with numbers.

If you are asking this because of a specific problem you have been given, it will be very helpful if you can quote the problem and tell us the context, so we can be sure what issues matter.

#### JeffM

##### Elite Member
If x^2 = √16, then how to calculate the value of 'x' in exponential form? I have done below in the root form
x^2 = √16, x= √(√16), x= √4 =2 Is the method is correct? if correct, please solve this problem in exponential form. I have tried below
x^2 = 16^1/2, x^2 = 4, x^2 = 2^2, x = 2 Am I correct? If correct, can you please solve it in any other methods?
Oh for goodness' sake, if you had given the problem completely and exactly originally, a great deal of time could have been saved.

$$\displaystyle x^2 = \sqrt{16} \implies x^2 = 16^{(1/2)} \implies (x^2)^{(1/2)}= \pm (16^{(1/2)})^{(1/2)} \implies$$

$$\displaystyle x^1 = \pm 16^{(1/4)} \implies x = \pm 2.$$

Moreover your solution is numerically incomplete: minus 2 is also a solution.

Last edited:

#### Subhotosh Khan

##### Super Moderator
Staff member
If x^2 = √16, then how to calculate the value of 'x' in exponential form? I have done below in the root form
x^2 = √16, x= √(√16), x= √4 =2 Is the method is correct? if correct, please solve this problem in exponential form. I have tried below
x^2 = 16^1/2, x^2 = 4, x^2 = 2^2, x = 2 Am I correct? If correct, can you please solve it in any other methods?
Are you sure that you are quoting the given problem EXACTLY - verbatim?
In the case it is - can you please explain what does "in exponential form" mean in the given context!

Better yet - give us the whole context - i.e. - where and how this problem was given to you?

Was the problem given to you in English - or are you translating?

#### mmm4444bot

##### Super Moderator
Staff member
x^2 = √y = y^(1/2)

so, x^2 = y^(1/2) Now What can I do?
When I suggested switching to exponential form, I was referring only to the simplification of sqrt(√y).

x^2 = √y

|x| = sqrt(√y)

x = ±[y^(1/2)]^(1/2)

x = ±y^(1/4)

In the last step, I used a property of exponents.

The given equation has been "solved for x".

The result shows the relationship between quantities x and y. It is also a formula for finding x, IF you already know a value for y.

EG:

Given that y = 2401, find all values of x.

x = ±2401^(1/4)

x = ±7​

#### mmm4444bot

##### Super Moderator
Staff member
… so that you don't have to remember anything important …
How's that? Did you intend to say, "so that you don't forget anything important"?