Simplifying $\sqrt[4]{161-72 \sqrt{5}}$

Question

$$\sqrt[4]{161-72 \sqrt{5}}$$

I tried to solve this as follows:

the resultant will be in the form of $a+b\sqrt{5}$ since 5 is a prime and has no other factors other than 1 and itself. Taking this expression to the 4th power gives $a^4+4 \sqrt{5} a^3 b+30 a^2 b^2+20 \sqrt{5} a b^3+25 b^4$. The integer parts of this must be equal to $161$ and the coeffecients of the roots must add to $-72$. You thus get the simultaneous system:

$$a^4+30 a^2 b^2+25 b^4=161$$ $$4 a^3 b+20 a b^3=-72$$

In an attempt to solve this, I first tried to factor stuff and rewrite it as:

$$\left(a^2+5 b^2\right)^2+10 (a b)^2=161$$ $$4 a b \left(a^2+5 b^2\right)=-72$$

Then letting $p = a^2 + 5b^2$ and $q = ab$ you get

$$4 p q=-72$$ $$p^2+10 q^2=161$$

However, solving this yields messy roots. Am I going on the right path?

Answer 1

$$\sqrt[4]{161-72\sqrt5}=\sqrt[4]{81-72\sqrt5+80}=\sqrt[4]{(9-4\sqrt{5})^2}=\sqrt{9-4\sqrt{5}}=\sqrt{4-4\sqrt{5}+5}=\sqrt{(2-\sqrt{5})^2}=\sqrt5-2$$ The trick is to notice that $72$ factors into $2*9*4$ and since $9^2+(4\sqrt5)^2=161$ you get this

Answer 2

Another approach. We can apply twice the following general algebraic identity involving nested radicals \begin{equation*} \sqrt{a-\sqrt{b}}=\sqrt{\frac{a+\sqrt{a^{2}-b}}{2}}-\sqrt{\frac{a-\sqrt{ a^{2}-b}}{2}}\tag{1} \end{equation*} to get \begin{equation*} \sqrt[4]{161-72\sqrt{5}}=\sqrt[4]{161- \sqrt{25\,920}}=\sqrt{5}-2. \end{equation*} The numerical computation can be carried out as follows:

\begin{eqnarray*} \sqrt[4]{161-72\sqrt{5}} &=&\left( \sqrt{\frac{161+\sqrt{161^{2}-25\,920}}{2} }-\sqrt{\frac{161-\sqrt{161^{2}-25\,920}}{2}}\right) ^{1/2} \\ &=&\left( \sqrt{\frac{161+1}{2}}-\sqrt{\frac{161-1}{2}}\right) ^{1/2} \\ &=&\sqrt{9-\sqrt{80}} \\ &=&\sqrt{\frac{9+\sqrt{9^{2}-80}}{2}}-\sqrt{\frac{9-\sqrt{9^{2}-80}}{2}} \\ &=&\sqrt{\frac{9+1}{2}}-\sqrt{\frac{9-1}{2}}\\ &=&\sqrt{5}-2. \end{eqnarray*}

ADDED. Note: If the radical were of the form $\sqrt{a+\sqrt{b}}$, then the applicable identity would be

\begin{equation*} \sqrt{a+\sqrt{b}}=\sqrt{\frac{a+\sqrt{a^{2}-b}}{2}}+\sqrt{\frac{a-\sqrt{ a^{2}-b}}{2}}.\tag{2} \end{equation*}

Proof (from Sebastião e Silva, Silva Paulo, Compêndio de Álgebra II, 1963). To find two rational numbers $x,y$ such that

\begin{equation*} \sqrt{a+\sqrt{b}}=\sqrt{x}+\sqrt{y},\text{ with }a,b\in \mathbb{Q}, \end{equation*}

we square both sides and rearrange the terms

\begin{equation*} 2\sqrt{xy}=a-x-y+\sqrt{b}. \end{equation*}

Squaring again yields \begin{equation*} 4xy=\left( a-x-y\right) ^{2}+2\left( a-x-y\right) \sqrt{b}+b. \end{equation*} Since $x,y\in \mathbb{Q}$, $a-x-y=0$, which means that $x,y$ satisfy the system of equations

\begin{equation*} x+y=a,\qquad xy=\frac{b}{4}. \end{equation*}

Consequently they are the roots of \begin{equation*} X^{2}-aX+\frac{b}{4}=0, \end{equation*}

i.e.

\begin{eqnarray*} x &=&X_{1}=\frac{a+\sqrt{a^{2}-b}}{2} \\ y &=&X_{2}=\frac{a-\sqrt{a^{2}-b}}{2}. \end{eqnarray*}

Answer 3

Denesting $\sqrt w = \sqrt{a+b\sqrt{n}}\,$ can be done by a simple formula that I discovered as a teenager. $ {\bf Simple\ Denesting\ Rule} \ \ \overbrace{\rm \color{#0a0}{subtract\ out}\ \sqrt{norm}^{\phantom .}}^{\textstyle\!\!\! w \to w - \sqrt{ww'} =:\, s\!\!\!\!\!\!}\!\!\!, \ {\rm then}\ \ \overbrace{\color{brown}{\rm divide\ out}\ \sqrt{{\rm trace}}^{\phantom .}}^{\textstyle s\,\to\, s/\sqrt{s+s'}\!\!\!\!\!\!\!}$ from that.

$\!\begin{align}{\rm Recall}\ \ w = a + b\sqrt{n}\rm \ \ has\ \ {\bf norm}\ &=\: w\:\cdot\: w' = (a + b\sqrt{n})\ \cdot\: (a - b\sqrt{n})\ =\: a^2\! - n\: b^2\\[4pt] {\rm and,\ furthermore,\ }w\rm \ \ has\ \ {\bf trace}\ &=\: w+w' = (a + b\sqrt{n}) + (a - b\sqrt{n})\: =\: 2\,a\end{align}$

In the norm/trace sqrts either sign works e.g. $\sqrt 1 = \pm1,\,$ so we choose what proves simplest.

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Here $\:161-72\sqrt 5\:$ has norm $= 1.\:$ $\rm\ \color{#0a0}{subtracting\ out}\ \sqrt{norm}\ = -1\ $ yields $\ 162-72\sqrt 5\:$

which has $\ {\rm\ \sqrt{trace}}\: =\: \sqrt{324}\ =\ 18.\ \ \ \ \rm \color{brown}{Dividing\ it\ out}\ \,$ of the above yields $\ \ \ 9-4\sqrt 5$

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Next $\:9-4\sqrt 5\:$ has norm $= 1.\:$ $\rm\ \color{#0a0}{subtracting\ out}\ \sqrt{norm}\ = 1\ $ yields $\ 8-4\sqrt 5\:$

with $\ {\rm\ \sqrt{trace}}\: =\: \sqrt{16}\ =\ 4.\ \ \ \ \rm \color{brown}{Dividing\ it\ out}\,\ $ of the above yields $\,\ \ \ 2-\sqrt 5$

Negating $\,2-\sqrt 5\,$ to get the positive square-root yields the sought result. We chose the signs in $\,\sqrt 1 = \pm 1$ so that arithmetic is simplest. Any choice will work as the proof below shows (e.g. we do both here). For many worked examples see prior posts on denesting. Below is a sketch of a proof.

Lemma $\ \ \sqrt w\, =\, \dfrac{s}t,\ \ \ \begin{align}s &\,=\, w \pm \sqrt{ww'}\\[.1em] t &\,=\: \pm\sqrt{s+s'}\end{align}\ $ when $\ \ \color{#90f}{\sqrt{ww'}\in\Bbb Q}$

Proof $\quad\ s^2 =\, w (w+w' \pm 2\sqrt{ww'})\, =\, w\, t^2$

Necessarily $\ \color{#90f}{\sqrt{ww'}\in \Bbb Q}\,$ if a denesting $\sqrt w = v = c + d\sqrt n\,$ exists, since

$$w = v^2\,\Rightarrow\, w' = v'^2\Rightarrow\, ww' = (vv')^2\in\Bbb Q^2\qquad$$

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Or $\rm \,\ x^2-t\:\!x+n^2 = 0\iff (t+2n)\,x = (x+n)^2\iff x = {\large \left[\frac{x+n}{\sqrt{t+2n}} \right]}^2$

Answer 4

I think this problem is a lot easier than the other answers would have one believe. Since $$161^2-5\cdot72^2=(161+72\sqrt5)(161-72\sqrt5)=1$$ We can see that $161+72\sqrt5$ is a unit in the ring of integers of $\mathbb{Q}(\sqrt5)$ Thus it must be $(\pm1)$ times a power of the fundamental unit, $\phi=\frac{1+\sqrt5}2$: $$161+72\sqrt5=\phi^n$$ Solving for $n$ we have $$n=\frac{\ln(161+72\sqrt5)}{\ln\left(\frac{1+\sqrt5}2\right)}=12$$ Thus $$(161+72\sqrt5)^{1/4}=\phi^3=2+\sqrt5$$

Answer 5

A (more complicated) approach that works on any nested radical, would be to use the Zippel Denesting Theorem.

$\sqrt[4]{161-72\sqrt{5}}$ is a fourth power exponent in $\mathbb{Q}(\sqrt{5})$ so setting the radical equal to its primitive root of unity and finding its roots gives us the simplification.

So we have: $\sqrt[4]{161-72\sqrt{5}}=x\iff x^4+72\sqrt{5}-161=0\iff (\sqrt{5}-2-x)(x+\sqrt{5}-2)(4\sqrt{5}-9-x^2)=0$ with the first one giving the correct denesting of $\sqrt{5}-2$.