Question

IF
$\sqrt{18 - 6 \sqrt{5} }$
=
$\sqrt{a } - \sqrt{b}$
, then prove a+b =18​

Answer 1

Given :

• √18 - 6√5 = √a - √b

To prove :

• a + b = 18

Solution :

~ It is given that √18 - √5 = √a - √b, We're asked to prove that a + b = 18, Let's use laws of exponents and solve it!

According to the question,

⟶ √18 - 6√5 = √a - √b

⟶ SOBS [ ∴ (a + b)² = a² + b² + 2ab ]

⟶ (√18 - 6√5)² = (√a - √b)²

⟶ 18 - 6√5 = (√a)² + (√b)² + 2√a√b

⟶ a + b + 2√a√b = 18 - 6√5

⟶ a + b + 2√a√b = 15 + 3 - 2√5√3

Comparing like terms,

⟶ a + b = 15 + 3

⟶ a + b = 18

∴ Hence, Proved!

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~ Additional information :

• a+ b)² = a² + b² + 2ab

• ( a - b )² = a² + b² - 2ab

• ( a + b )² + ( a - b)² = 2a² + 2b²

• ( a + b )² - ( a - b)² = 4ab

• ( a + b + c )² = a² + b² + c² + 2ab + 2bc + 2ca

• a² + b² = ( a + b)² - 2ab

• (a + b )³ = a³ + b³ + 3ab ( a + b)

• ( a - b)³ = a³ - b³ - 3ab ( a - b)

• If a + b + c = 0 then a³ + b³ + c³ = 3abc