Question

if m²+1/m²=34, find the value of m³+1/m³, using the positive value of m+1/m
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Answer 1

Step-by-step explanation:

GIVEN :-

$\sf \: {m}^{2} + \frac{1}{ {m}^{2} } = 34$

TO FIND :-

$\sf \: {m}^{3} + \frac{1}{ {m}^{3} }$

SOLUTION :-

$\sf \: \bigg(m + \frac{1}{m} \bigg) {}^{2} = {m}^{2} + \frac{1}{ { m}^{2} } + 2( \cancel{m})( \frac{1}{ \cancel{m}} ) \\ \\ \implies \sf \: \bigg(m + \frac{1}{m} \bigg) {}^{2} = 34 + 2 \\ \\ \implies\sf \: \bigg(m + \frac{1}{m} \bigg) {}^{2} = 36 \\ \\ \implies\sf \: \bigg(m + \frac{1}{m} \bigg) = \pm\sqrt{36} \\ \\ \implies\sf \: \bigg(m + \frac{1}{m} \bigg) = \pm6 \\ \\ \sf \: \: it \: is \: given \: that \: \: \sf \: \bigg(m + \frac{1}{m} \bigg) </p><p></p><p>\: \: is \: \: positive \: \\ \\ \sf \implies \: \sf \: \bigg(m + \frac{1}{m} \bigg) = 6$

$\sf \: Now \: cubing \: both \: sides \\ \\ \implies \sf \: \bigg(m + \frac{1}{m} \bigg) {}^{3} = {6}^{3} \\ \\ \implies \: \sf \: {m}^{3} + \frac{1}{ {m}^{3} } + 3(m)( \frac{1}{m} ) \bigg(m + \frac{1}{m} \bigg) = 216 \\ \\ \implies \: \sf \: {m}^{3} + \frac{1}{ {m}^{3} } + 3 \times (6) = 216 \\ \\ \sf \implies \: \: \sf \: {m}^{3} + \frac{1}{ {m}^{3} } + 18 = 216 \\ \\ \implies \: \sf \: {m}^{3} + \frac{1}{ {m}^{3} } = 216 - 18 \\ \\ \implies \: \sf \: {m}^{3} + \frac{1}{ {m}^{3} } = 198$

$\sf \: Hence \: \green{ \boxed{ {m}^{3} + \frac{1}{ {m}^{3} } = 198}}$