1.4 Operations on Real Numbers & Rationalization

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Step 1

Rationalize the first term: 7+353+5×3−53−5=21−75+95−159−5=6+254=3+52\frac{7+3\sqrt{5}}{3+\sqrt{5}} \times \frac{3-\sqrt{5}}{3-\sqrt{5}} = \frac{21 - 7\sqrt{5} + 9\sqrt{5} - 15}{9-5} = \frac{6 + 2\sqrt{5}}{4} = \frac{3+\sqrt{5}}{2}.

Step 2

Rationalize the second term: 7−353−5×3+53+5=21+75−95−159−5=6−254=3−52\frac{7-3\sqrt{5}}{3-\sqrt{5}} \times \frac{3+\sqrt{5}}{3+\sqrt{5}} = \frac{21 + 7\sqrt{5} - 9\sqrt{5} - 15}{9-5} = \frac{6 - 2\sqrt{5}}{4} = \frac{3-\sqrt{5}}{2}.

Step 3

Subtract: 3+52−3−52=(3+5)−(3−5)2=252=5\frac{3+\sqrt{5}}{2} - \frac{3-\sqrt{5}}{2} = \frac{(3+\sqrt{5}) - (3-\sqrt{5})}{2} = \frac{2\sqrt{5}}{2} = \sqrt{5}.

Final Answer

5\sqrt{5}

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