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# Algebra

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

Mar 7, 2024

#1
+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
+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

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

hairyberry Mar 8, 2024