Question

p/q+q/r+r/p=8 and q/p+r/q+p/r=11 then the value of p^3/q^3+q^3/r^3+r^3/p^3​

Answer 1

value of p³/q³ + q³/r³ + r³/p³ = 251.

given, p/q + q/r + r/p = 8 and q/p + r/q + p/r = 11

we have to find out p³/q³ + q³/r³ + r³/p³

let's trying formula,

a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca)

let p/q = a , q/r = b and r/p = c

then, p³/q³ + q³/r³ + r³/p³ - 3(p/q)(q/r)(r/p) = (p/q + q/r + r/p){p²/q² + q²/r² + r²/p² - (p/q)(q/r) - (q/r)(r/p) - (r/p)(p/q)}

⇒p³/q³ + q³/r³ + r³/p³ - 3 = (8){(p²/q² + q²/r² + r²/p²) - p/r - q/p - r/q}

using formula, a² + b² + c² = (a + b + c)² - 2(ab + bc + ca)

p²/q² + q²/r² + r²/p² = (p/q + q/r + r/p)² - 2(p/r + q/p + r/q)

⇒p³/q³ + q³/r³ + r³/p³ - 3 = 8{(8)² - 3 × 11}

⇒p³/q³ + q³/r³ + r³/p³ = 3 + 8 {64 - 33}

= 3 + 8 × 31

= 3 + 248

= 251