A Nice Algebra Problem | Math Olympiad | No Substitution

Hello everyone, you are welcome to how to evaluate this very nice radical expression root 52 over root 65 plus root 13 whole to the power 24. So let's start this root 52 can be written as root 13 times 5 plus this root 13 whole to the power this power 24. Next, this root 13 times 4 will become root 13 times root 4 divided by this root 13 times 5 will become root 13 times root 5 plus this root 13 whole to the power 24. Next, root 13 times this square root of 4 will become 2 divided by from the denominator we can factor out root 13. Root 13 as common factor in back left root 5 plus 1 whole to the power 24. Next, this root 13 will be cancelled out with this root 13 and this expression will become 2 divided by root 5 plus 1 whole to the power 24. Next, 2 divided by root 5 plus 1 whole to the power this 24 can be written as 3 times 8. Next, 2 over root 5 plus 1 whole cubed and whole to the power this power 8. Next, this 2 over root 5 plus 1 whole cubed will become 2 cubed divided by root 5 plus 1 whole cubed whole to the power 8. Next, this 2 cubed will become 8 and by using this algebraic identity A plus B whole cubed is equal to A cubed plus 3 times A squared B plus 3 times AB squared plus B cubed. This root 5 plus 1 whole cubed will become root 5 cubed plus 3 times root 5 squared times 1 plus 3 times root 5 times 1 squared plus 1 cubed and whole to the power 8. Next, 8 divided by this root 5 cubed will become 5 times root 5 plus this square will be cancelled out with this square root and 3 times 5 15 times 1 15 plus this 1 squared is 1 and 3 times 1 3 times this root 5 so this will become 3 times root 5 plus this 1 cubed is 1 and whole to the power 8. Next, 8 divided by this 5 times root 5 plus 3 times root 5 will become 8 times root 5 plus this 15 plus 1 16 whole to the power 8. Next, 8 divided by from the denominator we can factor out 8 in back left root 5 plus 2 whole to the power 8. This 8 will be cancelled out with this 8 and here we are left with 1 so this will become 1 divided by root 5 plus 2 whole to the power 8. Now, we rationalize the denominator so we write this 1 over root 5 plus 2 we multiply and divide by root 5 minus 2. We divide by root 5 minus 2 whole to the power 8. Next, one time root 5 minus 2 will become root 5 minus 2 divided by by using this algebraic identity A plus B times A minus B is equal to A squared minus B squared. This root 5 plus 2 times root 5 minus 2 will become root 5 squared minus 2 squared whole to the power 8. Next, root 5 minus 2 divided by this square will be cancelled out with this square root and here we are left with 5 minus 2 squared is 4 whole to the power 8. Next, root 5 minus 2 divided by 5 minus 4 is 1 whole to the power 8 and this will become root 5 minus 2 whole to the power 8. Now, this expression can be written as root 5 minus 2 whole squared and whole to the power 4. Next, by using this algebraic identity A minus B whole squared is equal to A squared plus B squared minus 2AB. This root 5 minus 2 whole squared will become root 5 squared plus 2 squared minus 2 times root 5 times 2 and whole to the power 4. Next, this square will be cancelled out with this square root and this 2 squared is 4. So, 5 plus 4 will become 9 minus 2 times 2 is 4 and this will become 4 times root 5 whole to the power 4. Next, this can be written as 9 minus 4 times root 5 whole squared whole to the power 2. Next, by using this algebraic identity this will become 9 squared plus 4 times root 5 squared minus 2 times 9 times 4 times root 5. Whole squared. Next, this 9 squared will become 81 plus this 4 squared is 16 and 16 times 5 will become 80 minus 2 times 9 18 times 4 72 times root 5 whole squared. Next, 81 plus 80 will become 161 minus 72 times root 5 whole squared. Now, this will become 161 squared plus 72 times root 5 squared minus 2 times 161 times 72 times root 5. This 161 whole squared can be written as 160 plus 1 whole squared plus this 72 times root 5 whole squared can be written as root 5 squared times 70 plus 2 squared minus this 2 times 161 will become 322 times this 72 times root 5. Next, this 160 plus 1 whole squared will become 160 squared plus 1 squared plus 2 times 160 times 1 plus this square will be considered with this square root and here we will act with 5 times this 70 plus 2 whole squared will become 70 squared plus 2 squared plus 2 times 70 times 2. We multiply this 322 by this 72. 322 multiplied by 72. 2 times 2, 4. 2 times 2, 4. 2 times 3, 6. And this 7 times 2, 14. 7 times 2, 14 plus 1, 15. 7 times 3, 21 plus 1, 22 and this will become 4, 8, 1, 3, 2. So, this will become minus 23184 times root 5. Next, this 16 squared is 256. We double the number of zeros. We write here 2 zeros plus 1 squared is 1 plus 2 times 160 times 1 will become 320 plus 5 times this 70 squared will become 4900 plus 2 squared will become 4 plus 2 times 70, 140 times 2, 280 minus 23184 times root 5. Now, we add these 3 numbers. So, this 25600 plus 1 plus 320 will become 25921 plus 5 times 4900 will become 24500 plus 5 times 4, 20 plus 5 times 280, 1400 minus this expression 23184 times root 5. Now, we add these 4 numbers 25921 and 24500, 14, 100 and 20. So, 1, 2 plus 2, 4, 9 plus 5, 14 plus 4, 18 and 5 plus 4, 9 plus 1, 10 plus 1, 11 and 2 plus 2, 4 plus 1, 5. So, this will become 51841 minus this expression 23184 times root 5. This is the simplest form. This is the final answer of this problem.