A Nice Radical Math Problem | Can You Solve This?

[@Shobhamaths]

a+2b=17;
a+b√5=161+(-72√5)
If u simplify by rationalizing
x=(√5-1) /2
x^12=161-72√5

[@nasrullahhusnan2289]

Let N=sqrt(45)+sqrt(49)
=7+3sqrt(5)
D=sqrt(125)+sqrr(121)
=11+5sqrt(5)
LHS=(N/D)¹² Note that 125-121=2²
Let C=11+5sqrt(5)
Multiply N/D by C/C:
NC/DC=¼[7+3sqt(5)][11-5sqrt(5)]
=¼[2-2sqrt(5)]
=½[1-sqrt(5)]
=-1/ß where ß=golden ratio
Thus LHS=-1/ß²⁴ and ß²⁴ can be easily get using Fibonacci

[@RashmiRay-c1y]

Let x=[√45 + √49]/[√125 + √121] = 1/2(√5-1). Thus, x^2=1-x. Hence, x^12 = 89-144x=161-72√5. So a=161 and b=-72. Thus, a+2b = 17.

[@akhildhawan5728]

17

[@SidneiMV]

(3√5 + 7)/(5√5 + 11)
= (3√5 + 7)(5√5 - 11)/4
= (75 - 33√5 + 35√5 - 77)/4
= (2√5 - 2)/4 = (√5 - 1)/2
[(√5 - 1)/2]¹² = a + b√5
x = (√5 - 1)/2 => a + b√5 = x¹²
x² = (3 - √5)/2
x² + x = 1 => x² = 1 - x
x³ = x - x² = x - (1 - x) => x³ = 2x - 1
x⁶ = 4x² - 4x + 1 = 4(1 - x) - 4x + 1
x⁶ = 5 - 8x
x¹² = 64x² - 80x + 25 = 64(1 - x) - 80x + 25
x¹² = 89 - 144x = 89 - 144(√5 - 1)/2
x¹² = a + b√5 = 161 - 72√5
a + 2b = 161 - 144 => a + 2b = 17

[@Fjfurufjdfjd]

17

[@ilhammaharramov9927]

17