Question

Can anyone help me solve this question?

e. Use the expansion of (1 + x)^5 to find the exact value of (1 + √5)^5

Answer 1

Answer:

  176 +80√5

Step-by-step explanation:

You want the exact value of (1 +√5)^5.

Binomial expansion

The coefficient of the expansion are found in Pascal's triangle (attached). Row 5 is used for the 5th power:

  (a +b)^5 = a^5 +5·a^4·b + 10·a^3·b^2 +10·a^2·b^3 +5·a·b^4 +b^5

Application

When a = 1 and b = √5, this becomes ...

  (1 +√5)^5 = 1 + 5·√5 +10·5 + 10·5√5 +5·25 +25√5

  = (1 +50 +125) +(5 +50 +25)√5

  (1 +√5)^5 = 176 +80√5

Answer 2

just a quick addition to "sqdancefan" superb reply above

$\qquad \qquad \textit{binomial theorem expansion} \\\\ \qquad \qquad (1+\sqrt{5})^{5}~\hspace{4em} \begin{array}{clcl} term&coefficient&value\\ \cline{1-3}&\\ 1&+1&(1)^{5 }(\sqrt{5})^0\\ 2&+5&(1)^{4}(\sqrt{5})^1\\ 3&+10&(1)^{3}(\sqrt{5})^2\\ 4&+10&(1)^{2}(\sqrt{5})^3\\ 5&+5&(1)^{1}(\sqrt{5})^4\\ 6&+1&(1)^{0}(\sqrt{5})^5 \end{array} \\\\[-0.35em] ~\dotfill$

$1(1)+5(1)(\sqrt{5})^1+10(\sqrt{5})^2+10(1)(\sqrt{5})^3+5(\sqrt{5})^4+(\sqrt{5})^5 \\\\\\ 1+5\sqrt{5}+50+50\sqrt{5}+125+25\sqrt{5}\implies {\Large \begin{array}{llll} 176+80\sqrt{5} \end{array}}$