Why is b^(m/n) equal to the mth power of the nth root of b?

  • Thread starter split
  • Start date
In summary, the formula b^(m/n) = (n√b)^m is true because of the laws of exponents and the definitions of negative and fractional exponents.
  • #1
split
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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.
 
Last edited:
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  • #2
b^(m/n)= nã(b^m) "the nth root of b to the m power"
you could also write (nãb)^m
Aaron
 
  • #3
I meant that but I was just thinking about too many things. It's been fixed. I'm asking for an explanation of why that is true.
 
  • #4


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

I don't know that it does. 31/2=(2[squ]1)1=2?
 
  • #5
The stupid errors just keep piling up don't they!

I take it you mean: "Why is bm/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. bn 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": bmbn= bm+n and (bm)n= bnm.

We then define bx 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 bn= bn+0= bnb0. As long as b is not 0 we can divide both sides of the equation by bn to bet b0= 1. That is, we MUST define b0= 1 or the laws of exponents will no longer hold.

Now we can see that bn+(-n)= b0= 1. If the laws of exponents are to hold for negative exponents as well, we must have bnb-n= bn+(-n)= 1 or, again dividing both sides of the equation by bn, b-n= 1/bn.

Finally, if (bm)n= bmn is to be true for all numbers, we must have (b1/m)m= b1= b. Since n &radic (b) is define as "the positive number whose nth power is b, we must define b1/m= m &radic (b).
 
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  • #6
Thanks HallsOfIvy, your explanation was very clear.

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.
 

FAQ: Why is b^(m/n) equal to the mth power of the nth root of b?

1. Why is the exponent of a fraction equivalent to the square root of the numerator and denominator?

The exponent of a fraction, or b^(m/n), represents the number of times the base (b) is multiplied by itself. When m and n are both integers, raising b to the power of m/n is equivalent to taking the nth root of b^m. This is because raising a number to a fractional power is the same as taking the root of that number by the denominator and then raising it by the numerator.

2. How does the property of fractional exponents relate to the laws of exponents?

The laws of exponents state that when multiplying numbers with the same base, their exponents can be added together. Similarly, when dividing numbers with the same base, their exponents can be subtracted. The property of fractional exponents follows these laws, as dividing a number by its nth root is equivalent to raising it to the power of 1/n. Therefore, b^(m/n) can be written as b^m * b^(-n), following the laws of exponents.

3. Are there any restrictions on the values of m and n in order for the equation to hold true?

Yes, m and n must both be integers and n cannot equal 0. This is because taking the root of a number by 0 is undefined, and taking the root of a negative number by an even root results in a complex number. Therefore, for the equation b^(m/n) = sqrt(n,m) to hold true, n must be a positive integer and m must be any integer.

4. Can this equation be applied to any base number?

Yes, this equation can be applied to any base number, as long as the restrictions on m and n are met. This property of fractional exponents applies to all numbers, whether they are integers, fractions, or irrational numbers. It is a fundamental rule of exponents that can be used for any base number.

5. How is the property of fractional exponents used in mathematical calculations?

The property of fractional exponents can be used to simplify and solve complex equations involving exponents. By converting fractional exponents to roots, mathematical calculations become easier to perform and understand. This property is also important in fields such as calculus, where fractional exponents are used to represent derivatives and integrals.

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