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Fill in the blanks with positive integers:
(3 + sqrt(5))^2*(2 - sqrt(5))^7 = ___ + ___ *sqrt(5)

 Mar 7, 2024

Best Answer 

 #1
avatar+394 
+3

\({(3+\sqrt{5})}^{2}*{(2-\sqrt{5})}^{7}\)

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 

\((14+6\sqrt{5})*(12238-5473\sqrt{5})=7142-3194\sqrt{5}\).

 Mar 8, 2024
 #1
avatar+394 
+3
Best Answer

\({(3+\sqrt{5})}^{2}*{(2-\sqrt{5})}^{7}\)

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 

\((14+6\sqrt{5})*(12238-5473\sqrt{5})=7142-3194\sqrt{5}\).

hairyberry Mar 8, 2024

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