A Nice Algebra Problem | Math Olympiad | Can you solve it?

hello welcome to how to solve this very nice square root math simplification problem square root of 81 divided by 9 minus square root of 80 first of all we make the denominator of this expression rational to make the denominator of this expression rational we multiply and divide this expression by the conjugate of this denominator like this square root of 81 divided by 9 minus square root of 80 we multiply this expression by the conjugate of this 9 minus square root of 80 conjugate of 9 minus square root of 80 is 9 plus square root of 80 and we divide this expression by 9 plus square root of 80 next square root of this 81 times 9 plus square root of 80 81 times 9 plus square root of 80 divided by by using this algebraic identity a minus b times a plus b is equal to a squared minus b squared this is 9 minus square root of 80 times 9 plus square root of 80 will become 9 squared minus square root of 80 squared next square root of 81 times 9 plus square root of 80 divided by 9 squared is 81 minus this square will be cancelled out with this square root and here left 80 next square root of 81 times 9 plus square root of 80 divided by 81 minus 81 next square root of this 81 times 9 plus square root of 80 divided by 1 is equal to 81 times 9 plus square root of 80 now by using this property square root of a times b is equal to square root of a times square root of b we can write this expression as square root of 81 times square root of 9 plus square root of 80 next square root of 81 is 9 times square root of this 9 plus square root of 80 can be written as 1 times 9 plus square root of 80 next 9 times square root of we can write this one as we can write 9 times this expression 9 plus square root of 80 next 9 times square root of now we multiply this 2 by this expression 9 plus square root of 80 2 times 9 plus square root of 80 divided by this 2 next 9 times square root of now this 2 times 9 18 plus this 2 times square root of 80 2 times square root of 80 divided by 2 next 9 times square root of now this 80 this 80 can be written as 10 times 8 and this 18 this 18 can be written as 10 plus 8 so we replace this 18 with 10 plus 8 plus 2 times square root of we replace this 80 with the 10 times 8 divided by 2 next 9 times square root of 10 plus 8 plus 2 times we can write this square root of 10 times 8 as square root of 10 times square root of 8 divided by 2 next square root of now we want to write this expression as a perfect square according to this algebraic identity a squared plus b squared plus 2ab is equal to a plus b whole squared in this expression we have this term square root of 10 as a and this term square root of 8 as b and we can write this 10 as we can write this 10 as square root of 10 squared we can write this 8 as we can write plus we write this 8 as square root of 8 squared plus 2 times square root of 10 times square root of 8 divided by this 2 next square root of now as this expression a squared plus b squared plus 2 a b is equal to a plus b whole squared this is square root of 10 squared plus square root of 8 squared plus 2 times square root of 10 times square root of 8 can be written as square root of 10 plus square root of 8 whole squared divided by 2 next by using this property square root of a over b is equal to square root of a over square root of b we can write this expression as square root of square root of 10 plus square root of 8 whole squared divided by square root of 2 next this square will be cancelled out with this square root and here left square root of 10 plus square root of 8 divided by square root of 2 and out of the square root we have 9 here 9 times this expression and 9 times this expression 9 times this expression so we write this expression times 9 next 9 times we can write this square root of 10 as square root of 5 times 2 plus we can write this square root of 8 as square root of 4 times 2 divided by square root of 2 next 9 times we can write this square root of 5 times 2 as square root of 5 times square root of 2 plus square root of 4 is 2 times square root of 2 divided by square root of 2 next 9 times now from here we can factor out square root of 2 square root of 2 as a common factor and in bracket left square root of 5 plus 2 divided by square root of 2 next 9 times this square root of 2 will be cancelled out with this square root of 2 and here left square root of 5 plus 2 next 9 times this to 18 plus 9 times square root of 5 9 times square root of 5 this is the simplest form this is the final answer.