Answer:
(a, b) = (9, 8) or (8, 9)
Step-by-step explanation:
Let the square of 17 + 6√8 be √a + √b that is,
[tex] \sqrt{17 + 6 \sqrt{8} } = \sqrt{a} + \sqrt{b} [/tex]
Square both sides
[tex]17 + 6 \sqrt{8} = {( \sqrt{a} + \sqrt{b} })^{2} [/tex]
Using the fact that (✓a + ✓b)² = a + b + 2✓ab
17 + 6√8 = a + b + 2√ab
This implies:
a + b = 17 ... (1) and
2√ab = 6√8 Divide both sides by 2
√ab = 3√8 Square both sides
ab = 9 × 8
ab = 72 ... (2)
From (1) b = 17 - a ... (3)
Substitute equation (3) in (2)
a ( 17 - a) = 72
17a - a² = 72
a² - 17a + 72 = 0 By Factorization
(a - 9)(a - 8) = 0
a = 9, a = 8
When a = 9
b = 17 - 9
b = 8
(a, b) = (9, 8)
When a = 8
b = 17 - 8
b = 9
(a, b) = (8, 9)
Therefore,
(a, b) = (9, 8) or (8, 9)
