If 3x+1/3x =3 find:
27x^3 + 1/27x^3
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[tex] \bf \underline{Given-} \\ [/tex]
[tex] \sf{3x + \frac{1}{3x} = 3 } \\ [/tex]
[tex] \bf \underline{To\: find-} \\ [/tex]
[tex] \sf{ the \:value \: of : \: {27x}^{3} + \frac{1}{ {27x}^{3} } = \: ?} \\ [/tex]
[tex] \bf \underline{Solution-} \\ [/tex]
[tex]\textsf{We have,}\\[/tex]
[tex] \sf{3x + \frac{1}{3x} = 3 } \\ [/tex]
[tex]\textsf{Cubing on both sides, we get}\\[/tex]
[tex] \sf{ \bigg(3x + \frac{1}{3x} \bigg)^{3} = (3 {)}^{3} } \\ [/tex]
[tex]\textsf{★Now, comparing this expression with (a+b)³, we get}\\[/tex]
[tex] \sf{ \: \: \: \: a = 3x \: and \: b = \frac{1}{3x}. } \\ [/tex]
[tex]★\textsf{Using identity (a+b)³=a³+b³+3ab(a+b),we get}\\[/tex]
[tex] \sf{ \bigg(3x + \frac{1}{3x} \bigg)^{3} = (3 {)}^{3} } \\ [/tex]
[tex] \sf{ \implies \: (3x {)}^{3} + \bigg(\frac{1}{3x} \bigg) ^{3} + 3(3x) \bigg( \frac{1}{3x} \bigg) \bigg(3x + \frac{1}{3x} \bigg) = 27} \\ [/tex]
[tex] \sf{ \implies \: 27x ^{3} + \frac{1}{27x ^{3} }+ 3(3x) \bigg( \frac{1}{3x} \bigg) \bigg(3x + \frac{1}{3x} \bigg) = 27} \\ [/tex]
[tex] \sf{ \implies \: 27x ^{3} + \frac{1}{27x ^{3} }+ 3( \cancel{3x}) \bigg( \frac{1}{ \cancel{3x}} \bigg) \bigg(3x + \frac{1}{3x} \bigg) = 27} \\ [/tex]
[tex] \sf{ \implies \: 27x ^{3} + \frac{1}{27x ^{3} }+ 3 \bigg(3x + \frac{1}{3x} \bigg) = 27} \\ [/tex]
[tex]\textsf{★Since 3x +$\frac{1}{3x}$ = 3 (Given)}\\[/tex]
[tex] \sf{ \implies \: 27x ^{3} + \frac{1}{27x ^{3} }+ 3 (3)= 27} \\ [/tex]
[tex] \sf{ \implies \: 27x ^{3} + \frac{1}{27x ^{3} } + 9= 27} \\ [/tex]
[tex] \sf{ \implies \: 27x ^{3} + \frac{1}{27x ^{3} } = 27 - 9} \\ [/tex]
[tex] \sf{ \implies \: 27x ^{3} + \frac{1}{27x ^{3} } = 18} \\ [/tex]
[tex] \bf \underline{Answer-} \\ [/tex]
[tex] \bf \underline{Hence, the\: value\:of : \: 27x ^{3} + \frac{1}{27x} ^{3} \: is \:18.} \\ [/tex]