SOLUTION: Given that {{{ sqrt( 6.4*10^n ) }}} is a rational number, what can you say about the value of n? What if both {{{ sqrt( 6.4*10^n ) }}} and {{{ root(3, 6.4*10^n ) }}} are rational?

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Question 398063: Given that is a rational number, what can you say about the value of n?
What if both and are rational? What can you say about the value of n?

Answer by Edwin McCravy(20055)   (Show Source): You can put this solution on YOUR website!
Given that is a rational number, what can you say about the value of n?













Only integer powers of 10 are rational. So the exponent
must be an integer, say k



n-1 = 2k

n = 2k+1

Therefore n must be odd in order that 
be rational.

What if both and are rational? What can you say about the value of n?

For












Only integer powers of 10 are rational. So the exponent
must be an integer, say p



n-1 = 3p

n = 3p+1

Therefore n must be 1 more than a multiple of 3

Now we set the exponents equal:

n = 3p+1 = 2k+1

3p = 2k

2p+p = 2k

p +  = 2k

 = 2k-p

The right side is an integer, so the left side must be also

Let that integer be A

Then 

 = A 

p = 2A

Substitute in

3p = 2k

3(2A) = 2k

6A = 2k

3A = k

Now supstitute in

n = 3p+1 = 2k+1

n = 3(2A)+1 = 2(3A)+1

n = 6A+1 = 6A+1 

So n must be 1 more than a multiple of 6.

Edwin

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