# What is (sqrt(5)+i)^6 in the form a+bi?

Oct 30, 2017

${\left(\sqrt{5} + i\right)}^{6} = - 176 + 56 \sqrt{5} i$

#### Explanation:

${\left(\sqrt{5} + i\right)}^{2} = {\left(\sqrt{5}\right)}^{2} + 2 \sqrt{5} i + {i}^{2} = 4 + 2 \sqrt{5} i$

${\left(4 + 2 \sqrt{5} i\right)}^{2} = {4}^{2} + 2 \left(4\right) \left(2 \sqrt{5} i\right) + {\left(2 \sqrt{5} i\right)}^{2}$

$\textcolor{w h i t e}{{\left(4 + 2 \sqrt{5} i\right)}^{2}} = 16 + 16 \sqrt{5} i - 20$

$\textcolor{w h i t e}{{\left(4 + 2 \sqrt{5} i\right)}^{2}} = - 4 + 16 \sqrt{5} i$

So:

${\left(\sqrt{5} + i\right)}^{6} = \left(4 + 2 \sqrt{5} i\right) \left(- 4 + 16 \sqrt{5} i\right)$

$\textcolor{w h i t e}{{\left(\sqrt{5} + i\right)}^{6}} = - 16 + 64 \sqrt{5} i - 8 \sqrt{5} i - 160$

$\textcolor{w h i t e}{{\left(\sqrt{5} + i\right)}^{6}} = - 176 + 56 \sqrt{5} i$

Check

We can make a rudimentary check of the answer by making sure that:

$\left\mid - 176 + 56 \sqrt{5} i \right\mid = {\left\mid \sqrt{5} + i \right\mid}^{6}$

We find:

$\left\mid \sqrt{5} + i \right\mid = \sqrt{{\left(\sqrt{5}\right)}^{2} + {1}^{2}} = \sqrt{5 + 1} = \sqrt{6}$

$\left\mid - 176 + 56 \sqrt{5} i \right\mid = \sqrt{{\left(- 176\right)}^{2} + {\left(56 \sqrt{5}\right)}^{2}} = \sqrt{30976 + 15680}$

$= \sqrt{46656} = \sqrt{{6}^{6}} = {\left(\sqrt{6}\right)}^{6}$

Oct 30, 2017

$- 176 + 56 \sqrt{5} i$

#### Explanation:

$6$th row of Pascal's triangle:

$1$ $6$ $15$ $20$ $15$ $6$ $1$

${\left(\sqrt{5} + i\right)}^{6} = {\left(\sqrt{5}\right)}^{6} + 6 \left({\left(\sqrt{5}\right)}^{5} \cdot i\right) + 15 \left({\left(\sqrt{5}\right)}^{4} \cdot {i}^{2}\right) + 20 \left({\left(\sqrt{5}\right)}^{3} \cdot {i}^{3}\right) + 15 \left({\left(\sqrt{5}\right)}^{2} \cdot {i}^{4}\right) + 6 \left(\left(\sqrt{5}\right) \cdot {i}^{5}\right) + {i}^{6}$

${\left(\sqrt{5}\right)}^{6} = {5}^{3} = 125$

$15 {\left(\sqrt{5}\right)}^{4} \cdot {i}^{2} = 15 \left({\left(\sqrt{5}\right)}^{4} \cdot - 1\right) = 15 \left({5}^{2} \cdot - 1\right) = 15 \cdot - 25 = - 375$

$15 {\left(\sqrt{5}\right)}^{2} \cdot {i}^{4} = 15 \cdot 5 \cdot 1 = 75$

${i}^{6} = {i}^{2} = - 1$

$125 - 375 + 75 - 1 = - 176$

${\left(\sqrt{5} + i\right)}^{6} = - 176 + 6 \left({\left(\sqrt{5}\right)}^{5} \cdot i\right) + 20 \left({\left(\sqrt{5}\right)}^{3} \cdot {i}^{3}\right) + 6 \left(\left(\sqrt{5}\right) \cdot {i}^{5}\right)$

$6 {\left(\sqrt{5}\right)}^{5} \cdot i = 6 \cdot 25 \sqrt{5} i = 150 \sqrt{5} i$

$20 {\left(\sqrt{5}\right)}^{3} \cdot {i}^{3} = 20 \cdot 5 \sqrt{5} \cdot - i = - 100 \sqrt{5} i$

$6 \left(\sqrt{5}\right) \cdot {i}^{5} = 6 \sqrt{5} \cdot i = 6 \sqrt{5} i$

$150 \sqrt{5} i - 100 \sqrt{5} i + 6 \sqrt{5} i = 56 \sqrt{5} \cdot i = 56 \sqrt{5} i$

${\left(\sqrt{5} + i\right)}^{6} = - 176 + 56 \sqrt{5} i$, in $a + b i$ form.