Question

If 2x + 3y =13 and xy=6 find the value of 8x3 + 27y3

Answer 1

Answer:

$\large{\underline{\boxed{\sf 8x^3 + 27y^3 = 793}}}$

Step-by-step explanation:

Given that ;

• 2x + 3y = 13
• xy = 6

Now, we have ;

$\sf 2x + 3y = 13$

On cubing both sides ;

$\implies \sf (2x + 3y) {}^{3} = (13) {}^{3} \\ \\ \implies \sf (2x) {}^{3} + {(3y)}^{3} + 3.2x.3y(2x + 3y) = 2197 \\ \\ \implies \sf 8x {}^{3} + 27y {}^{3} + 18xy(13) = 2197 \\ \\ \implies \sf 8x {}^{3} + 27y {}^{3} + 18(6)(13) = 2197 \\ \\ \implies \sf 8x {}^{3} + 27y {}^{3} + 1404 = 2197 \\ \\ \implies \sf 8x {}^{3} + 27y {}^{3} = 2197 - 1404 \\ \\ \boxed {\bf \therefore \: 8x {}^{3} + 27y {}^{3} = 793}$

Hence, the answer is 793.

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$\large{\underline{\underline{\mathfrak{\heartsuit \: Algebraic \: Identities : \: \heartsuit}}}}$

• (a - b)³ = a³ - b³ - 3ab (a - b)
• (a + b)³ = a³ + b³ + 3ab (a + b)
• (a + b)² = a² + 2ab + b²
• (a - b)² = a² - 2ab + b²
• (x + a)(x + b) = x² + x(a + b) + ab

Answer 2

Solution :-

Given that

2x + 3y = 13

xy = 6

Now as we know

(a + b)³ = a³ + b³ + 3ab(a + b)

→ a³ + b³ = (a + b)³ - 3ab(a + b)

Now using it in our question :-

Now as

(2x)³ = 8x³

(3y)³ = 27y³

So

→ (2x)³ + (3y)³ = (2x + 3y)³ - 3(2x)(3y)(2x + 3y)

→ (2x)³ + (3y)³ = (13)³ - 18xy(13)

→ (2x)³ + (3y)³ = 2197 - 234(6)

→ (2x)³ + (3y)³ = 2197 - 1404

→ (2x)³ + (3y)³ = 793

$\huge{\boxed{\sf{8x^3 + 27y^3 = 793}}}$