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C) First I simplified the equation√(23-6√6-4√2) =√a+√b to give 3+√2=√a+√b.
Tackling this equation is difficult in the sense that if we square both sides to remove the square roots we are left with 11= a +b and there are many positive pairs of numbers which can satisfy this equation. These include 1 and 10, 2 and 9, 3 and 8, 4 and 7, 5 and 6. Conversely, we can look at the equation in a different way by saying that 3=√a therefore meaning that a= 9 and similarly we can also say that √2=√b thus meaning b=2. In this way if we substitute in the values for both a and b, the equation is finally satisfied; should give an irrational value of 4.414213562 that continues in a non repeating pattern. To summarise, the positive integer values for a and b which would satisfy the equation √(23-6√6-4√2) =√a+√b are a=9 and b=2.