Question

if a2+1/a2=27 find the value of a3-1/a3​

Answer 1

given

${a}^{2} + \frac{1}{ {a}^{2} } = 27$

subtracting 2 from both sides

${a}^{2} + \frac{1}{ {a}^{2} } - 2 = 25 \\ \\ ( {a - \frac{1}{a} })^{2} = 25 \\ \\ a - \frac{1}{a} = 5 \: - - - (1)$

cubing both sides in equation (1)

$( {a - \frac{1}{a} })^{3} = {5}^{3} \\ \\ {a}^{3} - \frac{1}{ {a}^{3} } - 3(a - \frac{1}{a} )= 125 \\ \\ {a}^{3} - \frac{1}{ {a}^{3} } - 3 \times 5 = 125 \\ \\ {a}^{3} - \frac{1}{ {a}^{3} } - 15 = 125 \\ \\ {a}^{3} - \frac{1}{ {a}^{3} } = 140$