Question 1026549: Could someone please check my work on the following paragraph proof? This is to show why 11^1/n must equal n(sqrt)11 to uphold the properties of exponents, fill in the blanks to complete the paragraph proof. The answers I filled in are between ** Suppose b^n=11. Then b is the **nth** root of 11, which is written as **11^1/n**. Now consider that 11=11^1 and that n/n=1. By substitution, 11^1=11^n/n and by the **Power of a Power** property of exponents, 11^n/n=11^(1/n*n)=**(11^1/n)^n**. By the transitive property, since b^n=11 and 11=**(11^1/n)^n**, we know that b^n=**(11^1/n)^n)**. Therefore, b=11^1/n and as previously shown b=**n(sqrt)11**. Thus, **n(sqrt)11=11^1/n** Sorry it's so long, but I would appreciate any help!
Answer by robertb(5830)   (Show Source):
You can put this solution on YOUR website! At the very least I think you have to consider separating the proof by cases.
When n is even, it is trickier. For example,  doesn't mean  .
For n odd, it is automatic.
With that said, we have to impose the restriction that the root itself be a real number for the argument to hold. If complex roots are allowed then the notion of "roots" itself is not a single, particular number, but a class of (complex) numbers, and as such cannot be equal to just one number.
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