Question

37. Simply *. If 3x + 2y = 14 and xy = 8. find the value of 27x^3+ 8y^3​

Answer 1

Answer:

Coefficient of x⁶y³ is 672.

Step-by-step explanation:

General term of expansion (a+b)ⁿ is

\bf \: T_{r+1} = \: \: ^nC_r \: \: \large \frak{ a ^{n−r} b ^r}T

r+1

=

n

C

r

a

n−r

b

r

For (x+2y)⁹,

Putting n =9, a=x, b=2y

\begin{gathered} \bf \: T_{r+1} = \: \: ^{9} C_r (x) ^{9−r} (2y) ^r \\ \\ \bf \: T _{r+1} = \: \: ^{9} C_r (x) ^{9−r} .(y) ^r .(2) ^r\end{gathered}

T

r+1

=

9

C

r

(x)

9−r

(2y)

r

T

r+1

=

9

C

r

(x)

9−r

.(y)

r

.(2)

r

Comparing with x⁶ y³ , we get, r = 3

Therefore,

\begin{gathered} \bf \: T _{r+1} \\ \: \tt ^9C_3 (x)^9−3 .y³ .2³ \\ \tt\: \: 9! (2)³× x⁶ × y³) / (3!.6!) \\ \: \tt 672x⁶ y³\end{gathered}

T

r+1

9

C

3

(x)

9

−3.y³.2³

9!(2)³×x⁶×y³)/(3!.6!)

672x⁶y³

Answer 2

Answer:

2744 - 54x²y - 36xy²

Step-by-step explanation:

3x + 2y = 14

Cubing both sides

(3x+2y)³ = 14³

Using identity (a+b)³ = a³ + b³ + 3ab (a+b)

(3x)³ + (2y)³ + 3(3x)(2y) (3x+2y) = 2744

27x³ + 8y³ + 18xy (3x+2y) = 2744

27x³ + 8y³ + 54x²y + 36xy² = 2744

27x³ + 8y³ = 2744 - 54x²y - 36xy² (Answer)