Question

y +1/y= 4 then y^3-1/y^3=?​

Answer 1

Answer:

⇒ $y^3 - \frac{1}{y^3} = 30\sqrt{3}$

Step-by-step explanation:

Given :

$y+\frac{1}{y} = 4$

Find :

$y^3 - \frac{1}{y^3}$

Solution :

We know that,

$y+\frac{1}{y} = 4$

By squaring both the sides,

$y^2 + 2(y)(\frac{1}{y}) + (\frac{1}{y})^2 = (4)^2$

⇒ $y^2 + \frac{1}{y^2} + 2 = 16$

By subtracting 4 both the sides,

We get,

⇒ $y^2 + \frac{1}{y^2} + 2 - 4 = 16 - 4$

⇒ $y^2 + \frac{1}{y^2} - 2 = 12$

⇒ $y^2 + \frac{1}{y^2} - 2(y)(\frac{1}{y}) = 12$

⇒ $(y - \frac{1}{y})^2 = 12$

⇒ $y - \frac{1}{y} = \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$

⇒ $y - \frac{1}{y} = 2\sqrt{3}$

By cubing both the sides,

We get,

⇒ $(y - \frac{1}{y})^3 = (2\sqrt{3})^3$

⇒ $y^3 - 3(y)(\frac{1}{y})(y - \frac{1}{y}) - \frac{1}{y^3} = 24\sqrt{3}$

⇒ $y^3 - 3(y - \frac{1}{y}) - \frac{1}{y^3} = 24\sqrt{3}$

⇒ $y^3 - \frac{1}{y^3} = 24\sqrt{3} + 3(y - \frac{1}{y})$

⇒ $y^3 - \frac{1}{y^3} = 24\sqrt{3} + 3(2\sqrt{3})$

⇒ $y^3 - \frac{1}{y^3} = 24\sqrt{3} + 6\sqrt{3}$

⇒ $y^3 - \frac{1}{y^3} = 30\sqrt{3}$