Question

if 7 + root 5 / 7 minus root 5 minus 7 minus root 5 / 7 + root 5 is equal to a + b root 5 find the values of a and b

Answer 1

Heya !!!



7 + ✓5 / 7 - ✓5 - 7 - ✓5 / 7 + ✓5 = A + B✓5



LHS = 7 + ✓5/ 7 - ✓5 - 7 - ✓5 / 7 + ✓5




=> ( 7 + ✓5) ( 7 + ✓5) - ( 7 - ✓5 ) ( 7 - ✓5) / ( 7 + ✓5 ) ( 7 - ✓5)




=> (7+✓5)² - (7-✓5)² / (7)² - (✓5)²



=> (7)² + (✓5)² + 2 ×7 × ✓5 - { (7)² + (✓5)² - 2 × 7×✓5/ 49 - 5



=> 49 + 5 + 14✓5 - ( 49 + 5 - 14✓5) / 44



=> 49 + 5 + 14✓5 - 49 - 5 + 14✓5 / 44



=> 14✓5 + 14✓5/44



=> 28✓5/44



=> 4(7✓5) /44



=> 7✓5/11





Now,



LHS = RHS


7✓5 /11 = A + B✓5


Clearly,




A = 1 and B = 7/11




HOPE IT WILL HELP YOU...... :-)

Answer 2

(7+√5)/(7-√5) - (7-√5)/(7+√5)

=> [(7+√5)² - (7-√5)²]/[7²-(√5)²]

=> [ (7+5+14√5)- (14+5-14√2)]/[49-5]

=> [ 7+5+14√5-7-5+14√5]/[44]

=> 28√5/44

=> 7√5/11

Then,

7√5/11 = a + b√5

On comparing, we get,

a = 0
b = 7/11



I hope this will help you


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