If \(s-{1\over {s-8}}\) = 20, then the value of (s − 8)³ − \(1\over(s-8)^3\) is:

GIVEN:

s- 1/s-8 = 20 

FORMULA USED:

If x - 1/x = t 

Then x³ - 1/x³ = t³ + 3t 

Where t = constant 

CALCULATION:

Here, we have 

s- 1/s-8 = 20 

On subtracting -8 from the side we get

⇒ s -8 - 1/s-8 = 20 -8 

⇒ (s - 8 ) - 1/(s -8) = 12 

This is in the form of  x - 1/x = t 

Now, we know that x³ - 1/x³ = t³ + 3t 

⇒ ( s -8)³ - 1/( s -8)³ = 12³ + 3 × 12 

⇒ ( s - 8)³ - 1/( s -8)³ = 1728 + 36   = 1764 

Hence, the required value is 1764 .

Shortcut TrickS - 1/(S - 8) = 20

⇒ S - 1/(S - 8) - 8 = 20 - 8

⇒ (S - 8) - 1/(S - 8) = 12

Now, (S - 8)³ - 1/(S - 8)³ = [(S - 8) - 1/(S - 8)]³ + 3× (S - 8) × 1/(S - 8){(S - 8) - 1/(S - 8)} 

⇒ 12³ + 3 × 12 = 1728 + 36 = 1764