Question

If a + 1/a = 2, find a³ + 1/a³.​

Answer 1

Given→

$a + \frac{1}{a}$

To find→

${a}^{3} + \frac{1}{ {a}^{3} }$

Solution→

${a}^{2} + \frac{1}{ {a}^{2} } = ( a + \frac{1}{a} )^{2} - 2 \\ \\ \: \: \: \: = {2}^{2} - 2 \\ \: \: = 4 - 2 \\ = 2 \: \: \: \: \:$

${a}^{3} + \frac{1}{ {a}^{3} } = (a + \frac{1}{a} )( {a}^{2} + \frac{1}{ {a}^{2} } - 1) \\ \\ = 2(2 - 1) \: \: \: \: \: \: \: \: \: \: \: \\ \\ = 2(1) \: \: \: \: \: \: \: \:\: \: \: \: \: \: \: \: \:\:\\ \\ = 2\: \: \: \: \: \: \: \:\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$