Question

please answer please​

Answer 1

Answer:

The value of x³ + y³ is 970.

Step-by-step explanation:

Given :

x = (√3 - √2)/(√3 + √2) and

y = (√3 + √2)/(√3 - √2)

To find :

the value of x³ + y³

Solution :

First, find the value of x :

Rationalizing factor = (√3 - √2)

Multiply and divide the fraction by (√3 - √2)

  $\tt x=\dfrac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}} \times \dfrac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}} \\\\\\ \tt x=\dfrac{(\sqrt{3}-\sqrt{2})(\sqrt{3}-\sqrt{2})}{(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})}\\\\\\ \tt x=\dfrac{\sqrt{3}(\sqrt{3}-\sqrt{2})-\sqrt{2}(\sqrt{3}-\sqrt{2})}{\sqrt{3}(\sqrt{3}-\sqrt{2})+\sqrt{2}(\sqrt{3}-\sqrt{2})} \\\\\\ \tt x=\dfrac{\sqrt{3}^2-\sqrt{6}-\sqrt{6}+\sqrt{2}^2}{\sqrt{3}^2-\sqrt{6}+\sqrt{6}-\sqrt{2}^2} \\\\\\ \tt x=\dfrac{3-2\sqrt{6}+2}{3-2}\\\\ \tt x=\dfrac{5-2\sqrt{6}}{1}$

  x = 5 - 2√6

Find the value of y :

Rationalizing factor = (√3 + √2)

Multiply and divide the fraction by (√3 + √2)

  $\tt y=\dfrac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}} \times \dfrac{\sqrt{3}+\sqrt{2}}{\sqrt{3}+\sqrt{2}} \\\\\\ \tt y=\dfrac{(\sqrt{3}+\sqrt{2})(\sqrt{3}+\sqrt{2})}{(\sqrt{3}-\sqrt{2})(\sqrt{3}+\sqrt{2})}\\\\\\ \tt y=\dfrac{\sqrt{3}(\sqrt{3}+\sqrt{2})+\sqrt{2}(\sqrt{3}+\sqrt{2})}{\sqrt{3}(\sqrt{3}+\sqrt{2})-\sqrt{2}(\sqrt{3}+\sqrt{2})} \\\\\\ \tt y=\dfrac{\sqrt{3}^2+\sqrt{6}+\sqrt{6}+\sqrt{2}^2}{\sqrt{3}^2+\sqrt{6}-\sqrt{6}-\sqrt{2}^2} \\\\\\ \tt y=\dfrac{3+2\sqrt{6}+2}{3-2}\\\\ \tt y=\dfrac{5+2\sqrt{6}}{1}$

  y = 5 + 2√6

Formula :

a³ + b³ = (a + b)³ - 3ab (a + b)

Put a = x and b = y,

x³ + y³ = (x + y)³ - 3xy (x + y)

x³ + y³ = (5 - 2√6 + 5 + 2√6)³ - 3(5 - 2√6)(5 + 2√6) (5 - 2√6 + 5 + 2√6)

x³ + y³ = (5 + 5)³ - 3 [ 5² - (2√6)² ] (5 + 5)  | ∵ (a + b) (a - b) = a² - b² |

x³ + y³ = 10³ - 3 [ 25 - 4(√6)² ] (10)

x³ + y³ = 1000 - 3 [25 - 4(6)] (10)

x³ + y³ = 1000 - 3[25 - 24] (10)

x³ + y³ = 1000 - 3 (1) (10)

x³ + y³ = 1000 - 30

x³ + y³ = 970

The value of x³ + y³ is 970.