If frac{x^{24}+1}{x^{12}}=7,then find the val | vedclass

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(D) Given: $\frac{x^{24}+1}{x^{12}}=7$Dividing each term in the numerator by the denominator:$\frac{x^{24}}{x^{12}}+\frac{1}{x^{12}}=7$$x^{12}+\frac{1

Vedclass · Accepted Answer

$322$

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(D) Given: $\frac{x^{24}+1}{x^{12}}=7$Dividing each term in the numerator by the denominator:$\frac{x^{24}}{x^{12}}+\frac{1}{x^{12}}=7$$x^{12}+\frac{1}{x^{12}}=7$ ..... $(1)$We need to find the value of $\frac{x^{72}+1}{x^{36}}$,which is $x^{36}+\frac{1}{x^{36}}$.Using the identity $(a+b)^3 = a^3+b^3+3ab(a+b)$,let $a=x^{12}$ and $b=\frac{1}{x^{12}}$:$\left(x^{12}+\frac{1}{x^{12}}\right)^3 = (x^{12})^3 + \left(\frac{1}{x^{12}}\right)^3 + 3(x^{12})\left(\frac{1}{x^{12}}\right)\left(x^{12}+\frac{1}{x^{12}}\right)$Substitute the value from equation $(1)$:$7^3 = x^{36} + \frac{1}{x^{36}} + 3(1)(7)$$343 = x^{36} + \frac{1}{x^{36}} + 21$$x^{36} + \frac{1}{x^{36}} = 343 - 21$$x^{36} + \frac{1}{x^{36}} = 322$Thus,$\frac{x^{72}+1}{x^{36}} = 322$.

If $\frac{x^{24}+1}{x^{12}}=7$,then find the value of $\frac{x^{72}+1}{x^{36}}$.

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