+743 Fill in the blanks with positive integers:
(3 + sqrt(5))^2*(2 - sqrt(5))^7 = ___ + ___ *sqrt(5)
+399 \({(3+\sqrt{5})}^{2}*{(2-\sqrt{5})}^{7}\)
Simplify the left side first:
\({(3+\sqrt{5})}^{2}=14+6\sqrt{5}\)
Split some terms apart to find the right side.
\((2-\sqrt{5})^{2}=9-4\sqrt{5}\)
\({(2-\sqrt{5})}^{4}={(9-4\sqrt{5})}^{2}=161-72\sqrt{5}\)
\({(2-\sqrt{5})}^{7}={(2-\sqrt{5})}^{4}*{(2-\sqrt{5})}^{2}*{(2-\sqrt{5})}\).
\(=(161-72\sqrt{5})(9-4\sqrt{5})(2-\sqrt{5})=12238-5473\sqrt{5}\)
With a little help from calculators, this is
\((14+6\sqrt{5})*(12238-5473\sqrt{5})=7142-3194\sqrt{5}\).
+399 \({(3+\sqrt{5})}^{2}*{(2-\sqrt{5})}^{7}\)
Simplify the left side first:
\({(3+\sqrt{5})}^{2}=14+6\sqrt{5}\)
Split some terms apart to find the right side.
\((2-\sqrt{5})^{2}=9-4\sqrt{5}\)
\({(2-\sqrt{5})}^{4}={(9-4\sqrt{5})}^{2}=161-72\sqrt{5}\)
\({(2-\sqrt{5})}^{7}={(2-\sqrt{5})}^{4}*{(2-\sqrt{5})}^{2}*{(2-\sqrt{5})}\).
\(=(161-72\sqrt{5})(9-4\sqrt{5})(2-\sqrt{5})=12238-5473\sqrt{5}\)
With a little help from calculators, this is
\((14+6\sqrt{5})*(12238-5473\sqrt{5})=7142-3194\sqrt{5}\).
