Question

How do you rationalize the following denominator and reach a simplified solution?
Please show detailed steps.

Answer 1

\[(21-2\sqrt{21})\div(21+2\sqrt{21})\]

Answer 2

Multiply with conjugate pairs

Answer 3

(525-84root(21) ) / 357

Answer 4

One of my rough calcualtions

Answer 5

\[((21-2\sqrt{21})\times(21-2\sqrt{21})\div((21+2\sqrt{21})\times(21-2\sqrt{21}))\]\[(a-b)\times(a-b) = (a-b )^{2}\]
\[(a-b)\times(a+b) = a ^{2}-b ^{2}\]
\[(21-2\sqrt{21})^{2}\div(21^{2}-(2\sqrt{21})^{2})\]
\[(21^{2}+(2\sqrt{21})^{2}-2\times21\times2\sqrt{21})\div(21^{2}-(2\sqrt{21})^{2})\]
\[(441+4\times21-84\sqrt{21})\div(441-4\times21)\]
\[(525-84\sqrt{21})\div(357)\]
\[(25-4\sqrt{21})\div(17)\]

Answer 6

Excellent, thank you very much.