Root in Root

Algebra Level 3

If 91 + 2 2014 = a + b , a > b \sqrt{91 + 2\sqrt{2014}} = \sqrt{a} + \sqrt{b}, a > b .

What is the value of a − b a - b ?

Source : Mathematics Competition for Junior High School 2014, Forum Ilmiah Matematika Nasional Universitas Negeri Semarang .


The answer is 15.

Raihan Fauzan by Raihan Fauzan , 20, Indonesia

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1 solution

Raihan Fauzan
May 7, 2016

2014 = 53 × 38 , 91 = 53 + 38 2014 = 53 \times 38, 91 = 53 + 38

91 + 2 2014 = ( 53 + 38 ) + 2 ( 53 × 38 ) = ( 58 + 38 ) 2 = 53 + 38 , a > b \sqrt{91 + 2\sqrt{2014}} = \sqrt{(53 + 38) + 2\sqrt{(53 \times 38)}} = \sqrt{(\sqrt{58} + \sqrt{38})^{2}} = \sqrt{53} + \sqrt{38}, a > b

a = 53 , b = 38 a = 53, b = 38

Thus, a − b = 53 − 38 = 15 a - b = 53 - 38 = \boxed{15}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Yeahh the same way

I Gede Arya Raditya Parameswara - 4 years, 7 months ago

I request you to change 9 as 91. Thanks!

Samara Simha Reddy - 5 years, 1 month ago

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It happens again... My mistake. Thank you for reminding me again

Raihan Fauzan - 5 years, 1 month ago

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