Question

a = 7 + 4 root 3 , find the value of a^2 + 1 / a^2 .

Answer 1

Given :

a = 7 + 4√3

Find :

$\implies\:a^{2}\:+\:\dfrac{1}{a^{2}}$

Solution :

We have ..

$\Rightarrow\:a\:=\:7\:+\:4\sqrt{3}$

Now, squaring on both sides

$\Rightarrow\:a^{2}\:=\:(7\:+\:4\sqrt{3})^{2}$

(a + b)² = a² + b² + 2ab

$\Rightarrow\:a^{2}\:=\:(7)^{2}\:+\:(4\sqrt{3})^{2}\:+\:2(7)(4\sqrt{3}$

$\Rightarrow\:a^{2}\:=\:49\:+\:48\:+\:56\sqrt{3}$

$\Rightarrow\:a^{2}\:=\:97\:+\:56\sqrt{3}$

Now,

We have ..

$\implies\:a^{2}\:=\:97\:+\:56\sqrt{3}$

Take reciprocal of it.

$\Rightarrow\:\dfrac{1}{a^{2}}\:=\:\dfrac{1}{97\:+\:56\sqrt{3}}$

On rationalizing we get,

$\Rightarrow \: \dfrac{1}{ {a}^{2} } \: = \: \dfrac{97 \: - \: 56 \sqrt{3} }{1}$

$\implies\:\dfrac{1}{a^{2}}\:=\:97\:-\:56\sqrt{3}$

Now,

$\Rightarrow\:a^{2}\:+\:\dfrac{1}{a^{2}}\:=\:97\:+\:56\sqrt{3}\:+\:97\:-\:56\sqrt{3}$

$\implies\:a^{2}\:+\:\dfrac{1}{a^{2}}\:=\:194$

∴ a² + 1/a² = 194.

Answer 2

Answer:

$\huge{\pink{\green{\sf{\therefore a^{2}+\frac{1}{{a}^{2}}=194}}}}$

Step-by-step explanation:

$\huge{\pink{\green{\underline{\red{\sf{SOLUTION-}}}}}}$

• In the given question information given about value of a .

• We have to find the value that is given.

$\underline \bold{Given :} \\ \implies a = 7 + 4 \sqrt{3} \\ \\ \underline \bold{To \: Find : } \\ \implies {a}^{2} + \frac{1}{ {a}^{2} } = ?$

• According to given question :

$\bold{We\:know-} \\\bold{{a+b}^{2}=a^{2}+b^{2}+2ab}\\\bold{By\:using\:this\:formula-}\\\implies ( {a + \frac{1}{ a } })^{2} = {a}^{2} + \frac{1}{ {a}^{2} } + 2 \times a \times \frac{1}{a} \\ \implies (7 + 4 \sqrt{3} + \frac{1}{7 + 4 \sqrt{3} } )^{2} = {a}^{2} + \frac{1}{ {a}^{2} } + 2$

$\bold{Rationalise\:\frac{1}{7+4\sqrt{3}}}\\\implies (7 + 4 \sqrt{3} + \frac{1}{7 + 4 \sqrt{3} } \times \frac{7 - 4 \sqrt{3} }{7 - 4 \sqrt{3} } )^{2} = {a}^{2} + \frac{1}{ {a}^{2} } + 2 \\ \implies (7 + 4 \sqrt{3} + \frac{7 - 4 \sqrt{3} }{49 - 48} )^{2} = {a}^{2} + \frac{1}{ {a}^{2} } + 2 \\ \implies {a}^{2} + \frac{1}{ {a}^{2} } + 2 = (7 + 4 \sqrt{3} + 7 - 4\sqrt{3} )^{2} \\ \implies {a}^{2} + \frac{1}{ {a}^{2} } + 2 = {14}^{2} \\ \implies {a}^{2} + \frac{1}{ {a}^{2} } = 196 - 2 \\ \bold{\implies {a}^{2} + \frac{1}{ {a}^{2} } = 194}$