What are all the ordered pairs...?

Question

What are all the ordered pairs on integers (a,b), with 0<a<b, for which (a root 2007) (b root 2007) = (9 root 2007)?

(I don't mean 9 times sq root of 2007. It's square root of 2007 with a little a,b, or 9 on the left side inside the v part of the sq. root symbol. What is that called? Any help would be appreciated, thanks.)

Answer 1

The symbol means &quot;a-th&quot;, &quot;b-th&quot; or 9th root of 2007.

2007^(1/a) * 2007 ^ (1/b) = 2007^(1/9)

1/a + 1/b = 1/9

(b+a)/ab = 1/9

ab = 9(a+b)

b = 9a/(a-9)

a   b

10 90

12 36

18 18  (a = b, but why exclude it ?)

.

Answer 2

It&#039;s called the  nth root, where n is the number inside the &quot;v&quot; part.

You can also use this notation   2007 ^ (1/a) , where ^ indicates

exponentiation.

so you have  2007 ^(1/a) 2007^(1/b) = 2007 ^ (1/9)

using the rules of exponents you change this to

2007 ^ (1/a + 1/b) = 2007 ^ (1/9)

which is only true if

1/a + 1/b = 1/9

which you can arrange into

a + b = (ab)/9

This limits your search somewhat because ab/9 is going to

get bigger than a + b for larger values of a and b  . Also either a*b must be divisible by 9. Brute force determines that candidates meeting your criteria

are

(10,90) and (12,36)

(18,18) works in the equation, but doesn&#039;t satisfy the &quot;a&lt;b&quot; condition.

Answer 3

That little number in front is called the nth root. when you do the Sq root of a number there is an imaginary number 2 in front of the √ . if there is a 3 in front it is the 3rd root and so on. if your calculator can&#039;t do more than the Sq root. you can take the number you have and raise it to the fraction of that number

9√2007=2.3278...

2007^(1/9)=2.3278...

or 1/9=.11111111

so 2007^.11111111=2.3278...