Nested Radical

Nested radical adalah suatu bentuk akar yang di dalamnya terdapat bentuk akar lain. Contoh, \sqrt{5+2\sqrt5}

MENYEDERHANAKAN NESTED RADICAL

\sqrt{a+b\pm2\sqrt{ab}}=\sqrt{a}\pm\sqrt{b}

Sederhanakan bentuk akar berikut
\sqrt{3+2\sqrt2}

$\begin{array}{rl}\sqrt{3+2\sqrt{2}}& =\sqrt{(2+1)+2\sqrt{2\cdot 1}}\\ & =\sqrt{2}+\sqrt{1}\\ & =\sqrt{2}+1\end{array}$

Jadi, \sqrt{3+2\sqrt2}=\sqrt2+1

\sqrt{a\pm\sqrt{a\pm\sqrt{a\pm\cdots}}}=\dfrac12\left(\pm1+\sqrt{1+4a}\right)

Jika a=b(b-1), maka
\sqrt{a+\sqrt{a+\sqrt{a+\cdots}}}=b
\sqrt{a-\sqrt{a-\sqrt{a-\cdots}}}=b-1

Sederhanakan bentuk akar berikut
\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}

CARA 1

Misal x=\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}
$\begin{array}{rl}{x}^2& =2+\sqrt{2+\sqrt{2+\cdots }}\\ x^2& =2+x\\ x^2-x-2& =0\\ (x+1)(x-2)& =0\end{array}$
x=-1(TM) atau x=2

CARA 2

$\begin{array}{rl}2+\sqrt{2+\sqrt{2+\cdots }}& ={\displaystyle \frac{1}{2}}(1+\sqrt{1+4(2)})\\ & ={\displaystyle \frac{1}{2}}(1+\sqrt{9})\\ & ={\displaystyle \frac{1}{2}}(1+3)\\ & ={\displaystyle \frac{1}{2}}(4)\\ & =2\end{array}$

CARA 3

2=2\cdot1
maka,
\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}=2

Jadi, \sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}=2

\sqrt{1+a\sqrt{1+(a+1)\sqrt{1+(a+2)\sqrt{\cdots}}}}=a+1

\sqrt {ax+(n+a)^{2}+x{\sqrt {a(x+n)+(n+a)^{2}+(x+n){\sqrt {\mathrm {\cdots } }}}}}=x+n+a