Question

Simplify:

a.
`\sqrt{81+18√(7)+7 }`
b.
`\sqrt{25-10√(3)+3}`
c.
`\sqrt{29-12√(5)}`
d.
`\sqrt{8+2√(15)}`
Thank you!

Answer 1

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

`\textbf{b)}\\\\√(25 - 10 \sqrt 3 + 3)\\\\=√(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)}\\\\√(29-12\sqrt 5)\\\\=√(\left(2\sqrt 5 \right)^2-2\cdot 3\cdot 2\sqrt 5+3^2)\\\\=√(\left(2\sqrt 5 -3 \right)^2)\\\\=2\sqrt 5 - 3`

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