- #1

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**Why does b^(m/n) = (nsqrt(b))^m?**

Hi, as the subject says, why does b^(m/n) = (n√b)^m?

I don't understand how you can multiply a number by itself less than one times.

Thanks.

EDIT: Finally GOT IT RIGHT.

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- Thread starter split
- Start date

- #1

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Hi, as the subject says, why does b^(m/n) = (n√b)^m?

I don't understand how you can multiply a number by itself less than one times.

Thanks.

EDIT: Finally GOT IT RIGHT.

Last edited:

- #2

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b^(m/n)= nã(b^m) "the nth root of b to the m power"

you could also write (nãb)^m

Aaron

you could also write (nãb)^m

Aaron

- #3

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- #4

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I don't know that it does. 3Originally posted by split

Hi, as the subject says, why does b^(m/n) = (n√m)^m?

- #5

HallsofIvy

Science Advisor

Homework Helper

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The stupid errors just keep piling up don't they!

I take it you mean: "Why is b^{m/n}= n &radic (b)^{m}?"

Let's start with "I don't understand how you can multiply a number by itself less than one times."

You can't. b^{n} is defined as "multiply b by itself n times" only if n is a positive integer (counting number).

However, in that simple situation, we quickly derive the very useful "laws of exponents": b^{m}b^{n}= b^{m+n} and (b^{m})^{n}= b^{nm}.

We then define b^{x} for other number so that those laws remain true.

For example, IF the laws of exponents are to be true for x= 0, then we must have b^{n}= b^{n+0}= b^{n}b^{0}. As long as b is not 0 we can divide both sides of the equation by b^{n} to bet b^{0}= 1. That is, we MUST define b^{0}= 1 or the laws of exponents will no longer hold.

Now we can see that b^{n+(-n)}= b^{0}= 1. If the laws of exponents are to hold for negative exponents as well, we must have b^{n}b^{-n}= b^{n+(-n)}= 1 or, again dividing both sides of the equation by b^{n}, b^{-n}= 1/b^{n}.

Finally, if (b^{m})^{n}= b^{mn} is to be true for all numbers, we must have (b^{1/m})^{m}= b^{1}= b. Since n &radic (b) is define as "the positive number whose nth power is b, we must define b^{1/m}= m &radic (b).

I take it you mean: "Why is b

Let's start with "I don't understand how you can multiply a number by itself less than one times."

You can't. b

However, in that simple situation, we quickly derive the very useful "laws of exponents": b

We then define b

For example, IF the laws of exponents are to be true for x= 0, then we must have b

Now we can see that b

Finally, if (b

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- #6

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And yes, the errors kept piling up! I have fixed everything but the subject (I don't believe it can be changed. Am I wrong?) so if anyone wants to read it in the future it should make sense.

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