Question

Simplify:

Answer 1

$\textbf{a)}\\\\~~~\sqrt{81 + 18 \sqrt 7 +7}\\\\=\sqrt{9^2 +2\cdot 9\cdot \sqrt 7+ \left(\sqrt 7 \right)^2}\\\\=\sqrt{\left(9+\sqrt7 \right)^2}\\\\=9+\sqrt 7$

$\textbf{b)}\\\\\sqrt{25 - 10 \sqrt 3 + 3}\\\\=\sqrt{5^2 - 2 \cdot 5 \cdot \sqrt 3 + \left(\sqrt 3 \right)^2}\\\\=\sqrrt{\left(5 - \sqrt 3 \right)^2}\\\\=5-\sqrt 3$

$\textbf{c)}\\\\\sqrt{29-12\sqrt 5}\\\\=\sqrt{\left(2\sqrt 5 \right)^2-2\cdot 3\cdot 2\sqrt 5+3^2}\\\\=\sqrt{\left(2\sqrt 5 -3 \right)^2}\\\\=2\sqrt 5 - 3$

$\textbf{d)}\\\\\sqrt{ 8+ 2\sqrt{15}}\\\\=\sqrt{\left(\sqrt 5 \right)^2 + 2\cdot \sqrt{5 } \cdot \sqrt{3} +\left(\sqrt 3 \right)^2 }\\\\=\sqrt{\left(\sqrt 5 + \sqrt 3 \right)^2}\\\\=\sqrt 5 + \sqrt 3$