Question

If x2+1/x2=83 find the value of x3-1/x3

Answer 1

See the attach file for ur ans

Answer 2

The value of $\bf \red{{x}^{3} - \frac{1}{ {x}^{3} } = \pm756}$

Given:

• If ${x}^{2} + \frac{1}{ {x}^{2} } = 83$

To find:

• Find the value of ${x}^{3} - \frac{1}{ {x}^{3} }$

Solution:

Identity to be used:

1. $\bf {(a-b)}^{2}=a^2+b^2-2ab$
2. $\bf {(a-b)}^{3}=a^3-b^3-3ab(a-b)$

Step 1:

Subtract 2 from both sides of given equation.

${x}^{2} + \frac{1}{ {x}^{2} } - 2 = 83 - 2$

or

Write 2 as follows

${x}^{2} + \frac{1}{ {x}^{2} } - 2.x \frac{1}{x} = 81$

or

$( {x - \frac{1}{x} )}^{2} = ( {9)}^{2}$

or

taking square root both sides

$\bf x - \frac{1}{x} = \pm9...eq1$

Step 2:

Take cube of eq1,with +9

$( {x - \frac{1}{x}) }^{3} = 729$

open identity

${x}^{3} - \frac{1}{ {x}^{3} } - 3x \times \frac{1}{x} (x - \frac{1}{x} ) = 729$

or

${x}^{3} - \frac{1}{ {x}^{3} } - 3(9 ) = 729$

or

${x}^{3} - \frac{1}{ {x}^{3} } = 729 + 27$

or

$\bf {x}^{3} - \frac{1}{ {x}^{3} } = 756$

or

taking -9

${x}^{3} - \frac{1}{ {x}^{3} } = -729 - 27$

or

$\bf {x}^{3} - \frac{1}{ {x}^{3} } = -756$

Thus,

$\bf {x}^{3} - \frac{1}{ {x}^{3} } = \pm756$

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