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# Expand $(3+\sqrt{5})^4$. Enter your answer in the form $a + b\sqrt{5}$ for some positive integers $a$ and $b$.

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Expand $(3+\sqrt{5})^4$. Enter your answer in the form $a + b\sqrt{5}$ for some positive integers $a$ and $b$.

Jun 8, 2022

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$$\mathrm{Apply\:binomial\:theorem}:\quad \left(a+b\right)^n=\sum _{i=0}^n\binom{n}{i}a^{\left(n-i\right)}b^i$$

$$\sum _{i=0}^4\binom{4}{i}\cdot \:3^{\left(4-i\right)}\left(\sqrt{5}\right)^i$$

$$81+108\sqrt{5}+270+60\sqrt{5}+25$$

$$376+168\sqrt{5}$$

-Vinculum

Jun 8, 2022