Verified answer

[tex] \sqrt{52 - 10 \sqrt{27} } - \sqrt{52 + 10 \sqrt{27} } = \sqrt{25 - 10 \sqrt{27} + 27} - \sqrt{25 + 10 \sqrt{27} + 27 } = \sqrt{ {(5 - \sqrt{27} )}^{2} } - \sqrt{ {(5 + \sqrt{27} )}^{2} } = - 5 + \sqrt{27} - (5 + \sqrt{27} ) = - 5 + \sqrt{27} - 5 - \sqrt{27} = - 10[/tex]