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Answer 1

Qᴜᴇsᴛɪᴏɴ :-

if x + 2y = 8 and xy = 6, find the value of x³ + 8y³ = ?

Sᴏʟᴜᴛɪᴏɴ :-

→ x + 2y = 8

cubing both sides, we get,

→ (x + 2y)³ = 8³

using (a + b)³ = a³ + b³ + 3ab(a + b) in LHS, we get,

→ x³ + (2y)³ + 3 * x * 2y * (x + 2y) = 512

→ x³ + 8y³ + 6xy * (x + 2y) = 512

putting xy = 6 Now,

→ x³ + 8y³ + 6 * 6 * (x + 2y) = 512

putting (x + 2y) = 8 Now,

→ x³ + 8y³ + 36 * 8 = 512

→ x³ + 8y³ + 288 = 512

→ x³ + 8y³ = 512 - 288

→ x³ + 8y³ = 224 (Ans.)

Answer 2

${ \red{ \huge{ \underline{ \underline{ \mathfrak{Question:-}}}}}}$

▪ If x + 2y = 8 and xy = 6 , find the value of

${ \bold{ {x}^{ 3} + 8 {y}^{3} }}$

${ \red{ \huge{ \underline{ \underline{ \mathfrak{S olution: -}}}}}}$

the given equation is....

${ \bold{x + 2y = 8}}$

cubing both the sides....

${ \bold{ \implies{ {( x + 2y)}^{3} = {8}^{3} }}}$

using the identity of algebra ....

${ \boxed{ \red{ \bold{ {(a + b)}^{3} = {a}^{3} + {b}^{3} + 3ab(a + b)}}}}$

${ \bold{ \implies{ {(x + 2y)}^{3} = 512}}}$

${ \bold{ \implies{ {x}^{3} + {(2y)}^{3} + 3x(2y)(x + 2y) = 512}}}$

${ \bold{ \implies{ {x}^{3} + 8 {y}^{3} + 6xy(x + 2y) = 512}}}$

in the question, it's given that....

${ \pink{ \bold{xy = 6}}}$

${ \pink{ \bold{ x + 2y = 8}}}$

substituting the values of xy and (x + 2y) in the above equation......

${ \bold{ \implies{ {x}^{3} + 8 {y}^{3} + 6 \times 6 \times 8 = 512}}}$

${ \bold{ \implies{ {x}^{3} + 8 {y}^{3} = 512 - 288}}}$

${ \boxed{ \red{ \bold{ \implies{ {x}^{3} + 8 {y}^{3} = 288}}}}}$