Rationalize the numerator.

Question

((√a) -2)/(a-4)

Answer 1

( √a - 2 ) / ( a - 4 )

Multiply by the conjugate ( √a + 2 ) / ( √a + 2 )

( √a - 2 ) / ( a - 4 ) = ( √a - 2 ) ( √a + 2 ) / [ ( a - 4 ) ( √a + 2 ) ]

Difference of square formula:

x² - y² = ( x - y ) ( x + y )

So:

( √a - 2 ) ( √a + 2 ) = a - 4

( √a - 2 ) / ( a - 4 ) = ( a - 4 ) / [ ( a - 4 ) ( √a + 2 ) ] =

1 / ( √a + 2 )

--- OR ---

a - 4 = ( √a )² - 2²

Difference of square formula:

x² - y² = ( x - y ) ( x + y )

( √a )² - 2² = ( √a - 2 ) ( √a + 2 )

a - 4 = ( √a - 2 ) ( √a + 2 )

So:

( √a - 2 ) / ( a - 4 ) = ( √a - 2 ) / [ ( √a - 2 ) ( √a + 2 ) ] = 1 / ( √a + 2 )