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STEP I 2004 question 1 solutionTSR Wiki > Study Help > Subjects and Revision > Mathematics > STEP > STEP I 2004 question 1 solution Unparseable or potentially dangerous latex formula. Error 5: Image dimensions are out of bounds: 1127x22 .
(3 + 2 \sqrt{5} )^{3} \\ = (3 + 2 \sqrt{5})(3 + 2 \sqrt{5})^{2} \\ = (3 + 2 \sqrt{5})(9 + 12 \sqrt{5} + 20) \\ = (3 + 2 \sqrt{5})(29 + 12 \sqrt{5}) \\ = 87 + 36 \sqrt{5} + 58 \sqrt{5} + 120 \\ = 207 + 94 \sqrt{5}
At this moment in the solution it is helpful that
Hence, "c" being "3" seems like a sensible choice:
Hence:
Hence:
Now consider that:
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![(c - d \sqrt{2})^{3} \\ = (c - d \sqrt{2})(c - d \sqrt{2})^{2} \\ = (c - d \sqrt{2})((c^{2} + 2d^{2}) - 2cd \sqrt{2}) \\ = \left[ c(c^{2} + 6d^{2}) \right] - \left[ d(3c^{2} + 2d^{2} \right] \sqrt{2}](http://www.thestudentroom.co.uk/latexrender/pictures/78/783510314748f036dcbe2d4c97cad23f.png "(c - d \sqrt{2})^{3} \\ = (c - d \sqrt{2})(c - d \sqrt{2})^{2} \\ = (c - d \sqrt{2})((c^{2} + 2d^{2}) - 2cd \sqrt{2}) \\ = \left[ c(c^{2} + 6d^{2}) \right] - \left[ d(3c^{2} + 2d^{2} \right] \sqrt{2}")
![3 - 2 \sqrt{2} = \sqrt[3]{ 99 - 70 \sqrt{2} }](http://www.thestudentroom.co.uk/latexrender/pictures/cb/cbcf0109dd4c2f7dd7ad13831d06912a.png "3 - 2 \sqrt{2} = \sqrt[3]{ 99 - 70 \sqrt{2} }")
![(c + d \sqrt{2})^{3} = \left[ c(c^{2} + 6d^{2}) \right] + \left[ d(3c^{2} + 2d^{2} \right] \sqrt{2}](http://www.thestudentroom.co.uk/latexrender/pictures/70/701bca226a920feb5328f94830a39062.png "(c + d \sqrt{2})^{3} = \left[ c(c^{2} + 6d^{2}) \right] + \left[ d(3c^{2} + 2d^{2} \right] \sqrt{2}")
