Question

$a = 7 + \sqrt[4]{3 \: } them \\ n \: find \: the \: value \: of \: \sqrt{a} + \frac{1}{ \sqrt{a} }$
​

Answer 1

Correct Question :-

if a = 7 + 4√3 find the value of (√a - 1/√a) = ?

Sᴏʟᴜᴛɪᴏɴ :-

→ a = 7 + 4√3

Splitting the RHS Terms , we can write ,

→ a = 4 + 3 + 2 * 2 * √3

Or,

→ a = (2)² + (√3)² + 2 * 2 * √3

comparing it with a² + b² + 2ab = (a + b)² , we get,

→ a = (2 + √3)²

Square - Root Both sides Now, we get,

→ √a = (2 + √3).

Now,

→ 1/√a = 1/(2 + √3)

Rationalizing the RHS part Now,

→ 1/√a = 1/(2 + √3) * {(2 - √3) / (2 - √3)}

→ 1/√a = (2 - √3)/{(2 + √3)(2 - √3)}

using (a+b)(a-b) = a² - b² in Denominator,

→ 1/√a = (2 - √3) / (2² - √3²)

→ 1/√a = (2 - √3) / (4 - 3)

→ 1/√a = (2 - √3) .

Therefore,

→ (√a + 1/√a) = (2 + √3) + (2 - √3) = 4 (Ans.)

Answer 2

$\sf Correct\: Question :$

If vale of $\sf a = 7 + 4\sqrt{3}$ then find the value of the $\sf\sqrt{a}+ \frac{1}{ \sqrt{a}}$

⠀⠀⠀$\rule{160}{0.8}$

$\sf Answer :$

$:\implies\sf a = 7 + 4\sqrt{3}\\\\\\:\implies\sf a = 4 + 3 + 4 \sqrt{3}\\\\\\:\implies\sf a = {(2)}^{2} + {( \sqrt{3}) }^{2} + (2 \times 2 \times \sqrt{3})\\\\\\:\implies\sf a = (2 + \sqrt{3})^{2}\\\\\\:\implies\sf \sqrt{a} = 2 + \sqrt{3}\qquad...\:eq \:(l)$

⠀

$:\implies\sf \sqrt{a}=2+\sqrt{3}\\\\\\:\implies\sf \sqrt{a} = \sqrt{4} + \sqrt{3}\\\\{\scriptsize\qquad\bf{\dag}\:\:\textbf{As these are continuous number, hence}}\\\\:\implies\sf \dfrac{1}{ \sqrt{a} } = \sqrt{4} - \sqrt{3}\\\\\\:\implies\sf \dfrac{1}{ \sqrt{a} } =2 - \sqrt{3}\qquad...\:eq\:(ll)$

$\rule{180}{1.5}$

☯ $\underline{\boldsymbol{According\: to \:the\: Question :}}$

$\dashrightarrow\sf\:\:\sqrt{a} + \dfrac{1}{ \sqrt{a} }\\\\{\scriptsize\qquad\bf{\dag}\:\:\textbf{Putting value from eq. (l) \& eq. (ll)}}\\\\\dashrightarrow\sf\:\:(2 + \sqrt{3}) +(2 - \sqrt{3})\\\\\\\dashrightarrow\sf\:\:2 + \bcancel{\sqrt{3}} +2 - \bcancel{\sqrt{3}}\\\\\\\dashrightarrow\sf\:\:2 + 2\\\\\\\dashrightarrow\sf\:\:4$

⠀

$\therefore\:\underline{\textsf{Hence, required value will be \textbf{4}}}.$