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${\rm{Simplify, }}\sqrt {220 - 30\sqrt {35} } $

The given expression, 

${ = \sqrt {220 - 2 \times 15 \times \sqrt {35} } }$

${ = \sqrt {220 - 2 \times 3 \times 5 \times \sqrt 7  \times \sqrt 5 } }$

${ = \sqrt {\underbrace {{{(3\sqrt 5 )}^2}}_{9 \times 5 = 45} + \underbrace {{{(5\sqrt 7 )}^2}}_{25 \times 7 = 175} - 2(3\sqrt 5 )(5\sqrt 7 )} }$

${ = \sqrt {{{(3\sqrt 5  - 5\sqrt 7 )}^2}} }$

${ = |3\sqrt 5  - 5\sqrt 7 | = 5\sqrt 7  - 3\sqrt 5 }$

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