Question

If a^2+1/a^2=14 find the value of a^3+2/a^3.​

Answer 1

a² + $\dfrac{1}{ {a}^{2} }$ = 14

__________ [GIVEN]

We have to find a³ + $\dfrac{1*}{{a}^{3}}$

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Solution:

=> a² + $\dfrac{1}{ {a}^{2} }$ = 14

• Add 2 on both sides.

=> a² + $\dfrac{1}{ {a}^{2} }$ + 2 = 14 + 2

• We can also write this equal like..

=> a² + $\dfrac{1}{ {a}^{2} }$ + 2 $\dfrac{a}{a}$ = 16

We know that

a² + b² + 2ab = (a + b)²

So..

=> $({a \: + \: \dfrac{1}{a}) }^{2}$ = (4)²

=> a + $\dfrac{1}{a}$ = ${ (\sqrt{4}) }^{2}$

=> a + $\dfrac{1}{a}$ = 4

• Now cube on both sides.

=> $({a \: + \: \dfrac{1}{a}) }^{3}$ = (4)³

=> a³ + $\dfrac{1}{ {a}^{3} }$ + 3$(\dfrac{a}{a})$ $(a \: + \: \dfrac{1}{a})$ = 64

=> a³ + $\dfrac{1}{ {a}^{3} }$ + 3 $(a \: + \: \dfrac{1}{a} )$ = 64

=> a³ + $\dfrac{1}{ {a}^{3} }$ + 3(4) = 64

=> a³ + $\dfrac{1}{ {a}^{3} }$ = 64 - 12

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a³ + $\dfrac{1}{ {a}^{3} }$ = 52

__________ [ANSWER]