Solve the following Mathematical Problems

if m=7- $4 \sqrt{3}$ ,then find the value of $\sqrt{m} + \frac{1}{\sqrt{3}}$

Created: 1 year ago | Updated: 1 year ago

Updated: 1 year ago

$G i v e n m = 7 - 4 \sqrt{3} \Rightarrow \frac{1}{m} = \frac{1}{7 - 4 \sqrt{3}} = \frac{7 + 4 \sqrt{3}}{(7 + 4 \sqrt{3}) (7 - 4 \sqrt{3})} = \frac{7 + 4 \sqrt{3}}{{(7)}^{2} - {(4 \sqrt{3})}^{2}} = \frac{7 + 4 \sqrt{3}}{49 - 48} = 7 + 4 \sqrt{3} m + \frac{1}{m} = 7 - 4 \sqrt{3} + 7 + 4 \sqrt{3} = 14 ∴ m + \frac{1}{m} = 14 N o w, \sqrt{m} + \frac{1}{\sqrt{m}} \Rightarrow {(\sqrt{m} + \frac{1}{\sqrt{m}})}^{2} = {(\sqrt{m})}^{2} + 2 \times \sqrt{m} \times \frac{1}{\sqrt{m}} + {(\frac{1}{\sqrt{m}})}^{2} = m + 2 + \frac{1}{m} = m + \frac{1}{m} + 2 = 14 + 2 = 16 ∴ \sqrt{m} + \frac{1}{\sqrt{m}} = \sqrt{16} = 4$