Question

if m+n+p = 17 and mn+np+pm= 125, then find the value of m^(3) + n^(3) + p^(3) -3mnp

Answer 1

Given :

• m + n + p = 17

• mn + np + pm

To Find :

Value of

• m³ + n³ + p³ - 3mnp

Here,

a³ + b³ + c³ - 3abc = m³ + n³ + p³ - 3mnp

$\: \: \: \large \rm \: \therefore \: \: statisfies \: \: identity :$

Identity :

• a³ + b³ + c³ - 3abc

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

⇒ m³ + n³ + p³

= (m + n + p)(m² + n² + p² - mn - np - pm)

• Putting Values

$\large \rm \implies \: 17[{m}^{2} + {n}^{2} + {p}^{2} - (mn + np + pm)] \\ \\ \large \rm \implies \: 17[{m}^{2} + {n}^{2} + {p}^{2} - 125]$

Answer 2

Step-by-step explanation:

Given :

m + n + p = 17

mn + np + pm

To Find :

Value of

m³ + n³ + p³ - 3mnp

Here,

a³ + b³ + c³ - 3abc = m³ + n³ + p³ - 3mnp

\begin{gathered} \: \: \: \large \rm \: \therefore \: \: statisfies \: \: identity : \\ \end{gathered}

∴statisfiesidentity:

Identity :

a³ + b³ + c³ - 3abc

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

⇒ m³ + n³ + p³

= (m + n + p)(m² + n² + p² - mn - np - pm)

Putting Values

\begin{gathered} \large \rm \implies \: 17[{m}^{2} + {n}^{2} + {p}^{2} - (mn + np + pm)] \\ \\ \large \rm \implies \: 17[{m}^{2} + {n}^{2} + {p}^{2} - 125] \\ \\ \end{gathered}

⟹17[m

2

+n

2

+p

2

−(mn+np+pm)]

⟹17[m

2

+n

2

+p

2

−125]