1. Problem: Given $a = 19 + 8 \sqrt{3}$, find the value of $\sqrt{a} + \frac{1}{\sqrt{a}}$. 2. Formula and rules: To find $\sqrt{a} + \frac{1}{\sqrt{a}}$, we can use the identity: $$\left(\sqrt{a} + \frac{1}{\sqrt{a}}\right)^2 = a + 2 + \frac{1}{a}$$ This helps us find the value by first calculating $a + \frac{1}{a}$. 3. Calculate $\frac{1}{a}$: $$a = 19 + 8 \sqrt{3}$$ To find $\frac{1}{a}$, rationalize the denominator: $$\frac{1}{19 + 8 \sqrt{3}} = \frac{19 - 8 \sqrt{3}}{(19)^2 - (8 \sqrt{3})^2} = \frac{19 - 8 \sqrt{3}}{361 - 64 \times 3} = \frac{19 - 8 \sqrt{3}}{361 - 192} = \frac{19 - 8 \sqrt{3}}{169}$$ 4. Calculate $a + \frac{1}{a}$: $$a + \frac{1}{a} = (19 + 8 \sqrt{3}) + \frac{19 - 8 \sqrt{3}}{169} = \frac{(19 + 8 \sqrt{3}) \times 169}{169} + \frac{19 - 8 \sqrt{3}}{169} = \frac{(19 \times 169 + 8 \sqrt{3} \times 169) + 19 - 8 \sqrt{3}}{169}$$ Calculate numerator: $$19 \times 169 = 3211$$ $$8 \sqrt{3} \times 169 = 1352 \sqrt{3}$$ So numerator: $$3211 + 1352 \sqrt{3} + 19 - 8 \sqrt{3} = (3211 + 19) + (1352 \sqrt{3} - 8 \sqrt{3}) = 3230 + 1344 \sqrt{3}$$ Therefore: $$a + \frac{1}{a} = \frac{3230 + 1344 \sqrt{3}}{169}$$ 5. Now calculate: $$\left(\sqrt{a} + \frac{1}{\sqrt{a}}\right)^2 = a + 2 + \frac{1}{a} = \left(a + \frac{1}{a}\right) + 2 = \frac{3230 + 1344 \sqrt{3}}{169} + 2 = \frac{3230 + 1344 \sqrt{3} + 338}{169} = \frac{3568 + 1344 \sqrt{3}}{169}$$ 6. Simplify numerator and denominator: $$\frac{3568 + 1344 \sqrt{3}}{169} = \frac{16 \times 223 + 16 \times 84 \sqrt{3}}{169} = \frac{16 (223 + 84 \sqrt{3})}{169}$$ 7. Since $\left(\sqrt{a} + \frac{1}{\sqrt{a}}\right)^2 = \frac{16 (223 + 84 \sqrt{3})}{169}$, take the square root: $$\sqrt{\left(\sqrt{a} + \frac{1}{\sqrt{a}}\right)^2} = \sqrt{\frac{16 (223 + 84 \sqrt{3})}{169}} = \frac{4}{13} \sqrt{223 + 84 \sqrt{3}}$$ 8. To simplify $\sqrt{223 + 84 \sqrt{3}}$, assume it equals $m + n \sqrt{3}$: $$ (m + n \sqrt{3})^2 = m^2 + 2 m n \sqrt{3} + 3 n^2 = (m^2 + 3 n^2) + 2 m n \sqrt{3}$$ Equate: $$m^2 + 3 n^2 = 223$$ $$2 m n = 84 \Rightarrow m n = 42$$ Try $m=6$, $n=7$: $$6^2 + 3 \times 7^2 = 36 + 3 \times 49 = 36 + 147 = 183 \neq 223$$ Try $m=7$, $n=6$: $$7^2 + 3 \times 6^2 = 49 + 3 \times 36 = 49 + 108 = 157 \neq 223$$ Try $m=14$, $n=3$: $$14^2 + 3 \times 3^2 = 196 + 3 \times 9 = 196 + 27 = 223$$ $$m n = 14 \times 3 = 42$$ This works. 9. So: $$\sqrt{223 + 84 \sqrt{3}} = 14 + 3 \sqrt{3}$$ 10. Final answer: $$\sqrt{a} + \frac{1}{\sqrt{a}} = \frac{4}{13} (14 + 3 \sqrt{3}) = \frac{56 + 12 \sqrt{3}}{13}$$