Question

a² + 1/a² =
a³ + 1/a³ =
a + 1/a =
a³ - 1/a³=
a² - 1/a² =
answer these equations.
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Answer 1

Answer:

Question

Complete the equations :-

a² + 1/a² =

a³ + 1/a³ =

a + 1/a =

a³ - 1/a³=

a² - 1/a² =

Answer:-

Identity used :-

$\bf \: {(x + y)}^{2} = {x}^{2} + 2xy + {y}^{2}$

$\bf \: {(x - y)}^{2} = {x}^{2} - 2xy + {y}^{2}$

$\bf \: {x}^{2} - {y}^{2} = (x + y)(x - y)$

$\bf \: {x}^{3} + {y}^{3} = (x + y)( {x}^{2} - xy + {y}^{2} )$

$\bf \: {x}^{3} - {y}^{3} = (x - y)( {x}^{2} + xy + {y}^{2} )$

Solution:-

$\bf \:1. \: {a}^{2} + \dfrac{1}{ {a}^{2} } = {(a + \dfrac{1}{a}) }^{2} - 2 \times a \times \dfrac{1}{a}$

$\bf \:\: {a}^{2} + \dfrac{1}{ {a}^{2} } = {(a + \dfrac{1}{a}) }^{2} - 2$

$\bf \:2. \: {a}^{3} + \dfrac{1}{ {a}^{3} } = (a + \dfrac{1}{a} )( {a}^{2} - a \times \dfrac{1}{a} + \dfrac{1}{ {a}^{2} } )$

$\bf \:(a + \dfrac{1}{a} )( {a}^{2} -1 + \dfrac{1}{ {a}^{2} } )$

$\bf \:3. \: a + \dfrac{1}{a}$

$\bf \:4. \: {a}^{3} - \dfrac{1}{ {a}^{3} } = (a - \dfrac{1}{a} )( {a}^{2} + a \times \dfrac{1}{a} + \dfrac{1}{ {a}^{2} } )$

$\bf \: = (a - \dfrac{1}{a} )( {a}^{2} + 1+ \dfrac{1}{ {a}^{2} } )$

$\bf \:5. \: {a}^{2} - \dfrac{1}{ {a}^{2} } = (a + \dfrac{1}{a} )(a - \dfrac{1}{a} )$