#### stormbytes

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- Thread starter stormbytes
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The n-th root of a can be written as \(\displaystyle \sqrt[n]{a}\)=a^{1/n}. So if you have (-a)^{n}, where n is even, (-a)^{n}=a^{n}.

So (a^{n})^{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 (a^{2})^{1/2} \(\displaystyle \neq\) a.

So (a

The book gave you an example where (a

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

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 x

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 x

The square root of x

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

|x| = 4

That equation means

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

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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.The n-th root of a can be written as \(\displaystyle \sqrt[n]{a}\)=a^{1/n}. So if you have (-a)^{n}, where n is even, (-a)^{n}=a^{n}.

So (a^{n})^{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 (a^{2})^{1/2}\(\displaystyle \neq\) a.

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Thank you. That makes more sense.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 x^{2}= 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 x^{2}must be x.

The square root of x^{2}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

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

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

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

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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)Thank you. That makes more sense.