Question

If x − 1 /x = 1/3 , evaluate x^ 3 − 1 /x^ 3

Answer 1

EXPLANATION.

⇒ (x - 1/x) = 1/3.

As we know that,

Cubing on both sides of the equation, we get.

⇒ (x - 1/x)³ = (1/3)³.

As we know that,

Formula of :

⇒ (a - b)³ = a³ - 3a²b + 3ab² - b³.

Using this formula in the equation, we get.

⇒ (x)³ - 3(x)²(1/x) + 3(x)(1/x)² - (1/x)³ = (1/3)³.

⇒ (x)³ - 3x + 3/x - (1/x)³ = (1/3)³.

⇒ x³ - 3(x - 1/x) - 1/x³ = 1/27.

Put the values of (x - 1/x) = 1/3 in the equation, we get.

⇒ x³ - 3(1/3) - 1/x³ = 1/27.

⇒ x³ - 1 - 1/x³ = 1/27.

⇒ x³ - 1/x³ - 1 = 1/27.

⇒ x³ - 1/x³ = 1/27 + 1.

⇒ x³ - 1/x³ = (28/27).

Answer 2

Answer:

${\implies}{ \boxed{\mathbb{\red{given \: that }}}}\\ x - \frac{1}{x} = \frac{1}{3} \\ to \: find = {x}^{3} - \frac{1}{ {x}^{3} } \\ {\implies}{ \boxed{\mathbb{\red{solution}}}} \\ \\cubing \: both \: side \\ we \: get \\ (x - \frac{1}{x}) ^{3} =( \frac{1}{3} )^{3} \\ we \: know \: that \: identity \\ {\implies}{ \boxed{\mathbb{\pink{(a - b) {}^{3} = {a}^{3} - {b}^{3} - 3ab(a - b) }}}}\\ {x}^{3} - \frac{1}{ {x}^{3} } - 3 \times x \times \frac{1}{x} (x - \frac{1}{x} ) = \frac{1}{27} \\{\implies}{ \boxed{\mathbb{\blue{ putting \: value \: of \: x - \frac{1}{x} we \: get }}}}\\ {x}^{3} - \frac{1}{ {x}^{3} } - 3( \frac{1}{3} ) = \frac{1}{27} \\ {x}^{3} - \frac{1}{ {x}^{3} } - 1 = \frac{1}{27} \\ {x}^{3} - \frac{1}{ {x}^{3} } = \frac{1}{27} + 1 \\ {x}^{3} - \frac{1}{ {x}^{3} } = \frac{1 + 27}{27} \\ {\implies}{ \boxed{\mathbb{\orange{{x}^{3} - \frac{1}{ {x}^{3} } = \frac{28}{27} }}}}$

Step-by-step explanation:

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