Question

9. If a = 7 + 4√3 , then find the value of a²+1/a²​

Answer 1

$\huge\fbox\red{ \fbox\pink{Solutions:-}}$

$\fbox\red{ \fbox\orange{Given:-}}$

$\red{a = 7 + 4\sqrt{3} }$

$\fbox\red{ \fbox\blue{To find:-}}$

$\green {{a}^{2} + \frac{1}{ {a}^{2} } }$

$\fbox\red{ \fbox\green{By the problem:-}}$

$\red{a = 7 + 4\sqrt{3} } \\ = > \pink{\frac{1}{a} = \frac{1}{7 + 4\sqrt{3} } }$

$\fbox\red{ \fbox\red{Rationalise:-}}$

$\frac{1}{a} = \frac{1}{7 + 4\sqrt{3} } \\ = \frac{1 \times (7 - 4\sqrt{3}) }{(7 + 4 \sqrt{3})(7 - 4\sqrt{3} ) } \\ = \frac{7 - 4\sqrt{3} }{49 - 48} \\ = \frac{7 - 4\sqrt{3} }{1} \\ = 7 - 4 \sqrt{3}$

${a}^{2} + \frac{1}{ {a}^{2} } = \\ = > {a}^{2} + \frac{1}{ {a}^{2} } + 2 = 2 - - - - - (add \: 2 \: in \: both \: sides) \\ = > {(a)}^{2} + (\frac{1}{ {a}^{2} } ) + 2 \times a \times \frac{1}{a} = 2 \\ = > ( {a + \frac{1}{a}) }^{2} = 2$

$\fbox\red{ \fbox\red{Plot the value:-}}$

$({a + \frac{1}{{a}^{} } )}^{2} = 2 \\ = > {(7 + 4\sqrt{3} + 7 - 4\sqrt{3} )}^{2} = 2 \\ = > {(14)}^{2} = 2 \\ = > 196 - 2 \\ = > 194$

$\fbox\red{ \fbox\red{Answer:-}}$

194