Question

If 4x + 2y = 16 and xy = 8, find the value of 64x3 + 8y3​

Answer 1

$\textsf{\large{\underline{Solution}:}}$

Given That:

$\rm: \longmapsto4x + 2y= 16 - (i)$

$\rm: \longmapsto xy =8$

Cubing both sides of equation (i), we get:

$\rm: \longmapsto(4x + 2y)^{3} = {16}^{3}$

Using identity (a + b)³ = a³ + b³ + 3ab(a + b), we get:

$\rm: \longmapsto {64x}^{3} + 8y^{3} + 3 \times 4x \times 2y(4x + 2y)= {16}^{3}$

$\rm: \longmapsto {64x}^{3} + 8y^{3} +24xy(4x + 2y)= {16}^{3}$

Substitute the required values in the formula, we get:

$\rm: \longmapsto {64x}^{3} + 8y^{3} +24 \times 8 \times 16= {16}^{3}$

$\rm: \longmapsto {64x}^{3} + 8y^{3} +12\times 16\times 16= {16}^{3}$

$\rm: \longmapsto {64x}^{3} + 8y^{3}= {16}^{3} - 12 \times {16}^{2}$

$\rm: \longmapsto {64x}^{3} + 8y^{3}= {16}^{2}(16 - 12)$

$\rm: \longmapsto {64x}^{3} + 8y^{3}= 256 \times 4$

$\rm: \longmapsto {64x}^{3} + 8y^{3}=1024$

★ Which is our required answer.

$\textsf{\large{\underline{More To Know}:}}$

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