Having trouble understanding 'definition of nth root'

stormbytes

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Jul 19, 2020
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This is an excerpt from my book.
The second formula appears to be identical to the first, only stated in a more confusing fashion.
It can also be said that sq(16)^2 = |-2|^2.
What's the point of restating the exact same thing in a more complicated way??
Thank you!
 

yoscar04

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Jun 3, 2020
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The n-th root of a can be written as \(\displaystyle \sqrt[n]{a}\)=a1/n. So if you have (-a)n, where n is even, (-a)n=an.
So (an)1/n=|a|. Try and check what happen when n is not even, i.e. ((-3)3)1/3=?
The book gave you an example where (a2)1/2 \(\displaystyle \neq\) a.
 
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Otis

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What's the point of restating the exact same thing …
Hi stormbytes. The two situations are not exactly the same.

The point they're making is this: When a negative number has been squared, we loose the original sign. In other words, if I give you the number 16, then you can't know what number was squared to obtain that 16.

Here's an example that shows a common beginner's mistake:

Solve the equation x2 = 16

The student takes the square root of each side and writes

x = 4

They missed the solution x = -4 because they incorrectly assumed that the square root of x2 must be x.

The square root of x2 is not simply x. We need to express it as |x| because there's two possibilities.

Therefore, when we take the square root of each side, we write

|x| = 4

That equation means

x = 4 \(\quad\) OR \(\quad\) x = -4

😎
 

stormbytes

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The n-th root of a can be written as \(\displaystyle \sqrt[n]{a}\)=a1/n. So if you have (-a)n, where n is even, (-a)n=an.
So (an)1/n=|a|. Try and check what happen when n is not even, i.e. ((-3)3)1/3=?
The book gave you an example where (a2)1/2 \(\displaystyle \neq\) a.
Thank you. I’ll revisit your answer once I’ve done radical exponents. Will have a better sense of what you are illustrating in your examples.
 

stormbytes

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Jul 19, 2020
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Hi stormbytes. The two situations are not exactly the same.

The point they're making is this: When a negative number has been squared, we loose the original sign. In other words, if I give you the number 16, then you can't know what number was squared to obtain that 16.

Here's an example that shows a common beginner's mistake:

Solve the equation x2 = 16

The student takes the square root of each side and writes

x = 4

They missed the solution x = -4 because they incorrectly assumed that the square root of x2 must be x.

The square root of x2 is not simply x. We need to express it as |x| because there's two possibilities.

Therefore, when we take the square root of each side, we write

|x| = 4

That equation means

x = 4 \(\quad\) OR \(\quad\) x = -4

😎
Thank you. That makes more sense.
 

JeffM

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Sep 14, 2012
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5,435
The laws of exponents are usually expressed as follows

\(\displaystyle \text{Given } a \text { is a real number } > 0 \text { then stuff}\)

This is done to avoid complications that arise when a is not a positive real number.

One major complication is that if \(\displaystyle a^{2n}\) is a positive real number, then

\(\displaystyle \exists \ b < 0 \text { such that } b^2 = a^{2n} \text { and } \exists \ c > 0 \text { such that } c^2 = a^{2n}.\)

Of course, it can easily be shown thar b = - c, but still we would like to know whether we are dealing with a negative number or not. Therefore we define

\(\displaystyle \sqrt{u^2} = |u|.\)

How then do we describe the other one?

\(\displaystyle - \sqrt{u^2} = -|u|.\)

Notice that

\(\displaystyle (-|u|)^2 = (-1)^2(|u|)^2 = 1 * (|u|)^2 = (|u|)^2.\)
 

babadany2999

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Jun 9, 2020
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Thank you. That makes more sense.
Let a < 0 => sqrt(a^2) > 0 -correct , now if you do sqrt(a^2)=a ,then a < 0 which would be incorrect since you would be squaring it inside the square root(the correct way) or just say sqrt(a^2) = abs(a)
 
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