Can Higher Degree Nested Radicals Be Simplified?

[FranzS]

TL;DR Summary

How to solve nested radicals of higher degree

Hi PF.
I'm aware of a formula for solving (when possibile) nested square roots of the type $\sqrt{a+\sqrt{b}}$.
But is there any formula/strategy for solving higher degree nested radicals? For example, I cannot understand how one can solve...
$$
\sqrt[5]{\frac{5 \sqrt{5}-11}{2^6}}=\frac{1}{4} \left( \sqrt{5}-1 \right)
$$

 

[Mark44]

TL;DR Summary: How to solve nested radicals of higher degree

For example, I cannot understand how one can solve... $$
\sqrt[5]{\frac{5 \sqrt{5}-11}{2^6}}=\frac{1}{4} \left( \sqrt{5}-1 \right)
$$

This is not an equation that you would "solve." Unless this is just something that you made up, it looks to me like the left side simplifies somehow to what's on the right side.

 

[FranzS]

Yep, that's the loose meaning of "solve" in this case.

 

[Frabjous]

One would start by raising both sides to the fifth power.

 

[FranzS]

One would start by raising both sides to the fifth power.

Ok, but I would like to get the simplified expression from the one with the fifth root, not just prooving they're equal

 

[Mark44]

Yep, that's the loose meaning of "solve" in this case.

Very loose, as the usual meaning is to find one or more values for an unknown variable. As suggested by @Frabjous, a good start would be to raise each side to the 5th power. If you eventually arrive at an equation with both sides being the same, using reversible operations, that would indicate that the first equation was an identity. IOW, that the expression on the left side can be simplified to what is shown on the right side.

 

[Vanadium 50]

TL;DR Summary: How to solve nested radicals of higher degree

For example, I cannot understand how one can solve...

Are you sure it is correct? I plugged in the LHS and RHS and get different numbers.

 

[FranzS]

Are you sure it is correct? I plugged in the LHS and RHS and get different numbers.

Yes, it's correct!

 

[FactChecker]

Are you sure it is correct? I plugged in the LHS and RHS and get different numbers.

I got that they are equal (to the accuracy of my calculations).

 

[mcastillo356]

Hi @FranzS

One would start by raising both sides to the fifth power.

Here you got the first nice step. As you know, we can rewrite the expression on the left side this way:

$$\displaystyle\frac{(5\sqrt{5}-11)^{1/5}}{2^{6/5}}$$

These are Exponent Rules, for $a\neq{0}$ and $b\neq{0}$

Product Rule

$$ a^{x}\times{a^{y}}=a^{x+y}$$

Quotient Rule

$$a^{x}/a^{y}=a^{x-y}$$

Power Rule

$$\Big(a^{x}\Big)^{y}=a^{x\,y}$$

Power of a Product Rule

$$(ab)^{x}=a^{x}b^{x}$$

Power of a Fraction Rule

$$\Big(\frac{a}{b}\Big)^{x}=\displaystyle\frac{a^x}{b^x}$$

Zero exponent

$$a^0=1$$

Negative Exponent

$$a^{-x}=\displaystyle\frac{1}{a^x}$$

Fractional Exponent

$$a^{\frac{x}{y}}=\sqrt[y]{a^x}$$

Does it help?

Best wishes!

 

[SammyS]

@mcastillo356 ,
I'm sure what @Frabjous means by "raising both sides to the fifth power" is to use an exponent of $5$, not an exponent of $\dfrac 1 5$.

 

[mcastillo356]

Hi,. @SammyS, PF, I'm not native. Probably that's why my post might have been misleading.
My intention has been to write down the OP's equality's left side as an exponential. And I've also been googling the rules to work out that way.
Please, feel free to edit or erase my message.
Best wishes!

 

[renormalize]

TL;DR Summary: How to solve nested radicals of higher degree

As I interpret your question, given the radical $R\equiv\sqrt[5]{\frac{5\sqrt{5}-11}{2^{6}}}$, you seek to actually derive the simplified form $R=\frac{1}{4}\left(\sqrt{5}-1\right)$, having no prior knowledge of that expression. That's actually easy to accomplish by making use of the conjugate-element $\overline{R}$ of $R$:$$R=\left(\frac{5\sqrt{5}-11}{2^{6}}\right)^{\frac{1}{5}},\quad\overline{R}=\left(\frac{5\sqrt{5}+11}{2^{6}}\right)^{\frac{1}{5}}\tag{1a,1b}$$from which it follows that $R\overline{R}=\frac{1}{4}$. Then, noting that the simple product $\left(\sqrt{5}-1\right)\left(\sqrt{5}+1\right)=4$, we arrive at the identity:$$R\overline{R}=\frac{1}{4}=\left(\frac{5\sqrt{5}-11}{2^{6}}\right)^{\frac{1}{5}}\left(\frac{5\sqrt{5}+11}{2^{6}}\right)^{\frac{1}{5}}\equiv\left(\frac{\sqrt{5}-1}{4}\right)\left(\frac{\sqrt{5}+1}{4}\right)\tag{2}$$Finally, the most profound observation is that we may individually equate the elements and their conjugates in eq.(2) to develop the two separate identities:$$R=\left(\frac{5\sqrt{5}-11}{2^{6}}\right)^{\frac{1}{5}}\equiv\frac{\sqrt{5}-1}{4},\quad\overline{R}=\left(\frac{5\sqrt{5}+11}{2^{6}}\right)^{\frac{1}{5}}\equiv\frac{\sqrt{5}+1}{4}\tag{3a,3b}$$which both addresses your specific question and yields a free bonus identity!
(Note: I'm no mathematician, but to my understanding, the reason the last factorization step is true stems from Galois theory. Paging @fresh_42?)

 

[Frabjous]

@mcastillo356 ,
I'm sure what @Frabjous means by "raising both sides to the fifth power" is to use an exponent of $5$, not an exponent of $\dfrac 1 5$.

$\sqrt[5]\frac{5\sqrt5-11} {2^6}={\frac 1 4}(\sqrt5-1)$
$\frac{5\sqrt5-11} {2^6}={\frac 1 {2^{10}}}(\sqrt5-1)^5$
$16(5\sqrt5-11) =(\sqrt5-1)^5$
Now it is just simple algebra. Notice that raising $\sqrt5$ to an integer power gives either an integer or an integer multiple of $\sqrt5$

 

[FactChecker]

Finally, the most profound observation is that we may individually equate the elements and their conjugates in eq.(2) to develop the two separate identities:$$R=\left(\frac{5\sqrt{5}-11}{2^{6}}\right)^{\frac{1}{5}}\equiv\frac{\sqrt{5}-1}{4},\quad\overline{R}=\left(\frac{5\sqrt{5}+11}{2^{6}}\right)^{\frac{1}{5}}\equiv\frac{\sqrt{5}+1}{4}\tag{3a,3b}$$which both addresses your specific question and yields a free bonus identity!
(Note: I'm no mathematician, but to my understanding, the reason the last factorization step is true stems from Galois theory. Paging @fresh_42?)

Interesting! I would like to know more about this step.

 

[fresh_42]

Paging @fresh_42?)

Proving the identity was never the problem. I'm sure what @FranzS meant was:
$$
0=x^5-\dfrac{5\sqrt{5}-11}{64} \Longrightarrow x=\dfrac{1}{4}\left(\sqrt{5}-1\right) \text{ is one (of 5) solution(s) }
$$
How can we figure out how this root is found?

I tried several methods (your conjugation trick, polynomial division, an ansatz, complex numbers, minimal polynomial ...) but couldn't find one that succeeded. I think meanwhile that @FranzS has taken $\dfrac{\sqrt{5}-1}{4}$ to WA, asked it for $\left(\dfrac{\sqrt{5}-1}{4}\right)^5,$ and posted this question. I finally gave up since I saw no general value in the question: "Does $x^5-\dfrac{5\sqrt{5}-11}{64}\in \mathbb{Q}(\sqrt{5})[x]$ has a root?" when there are technical tools like WA which easily answer that question.

 

[mcastillo356]

$\sqrt[5]\frac{5\sqrt5-11} {2^6}={\frac 1 4}(\sqrt5-1)$
$\frac{5\sqrt5-11} {2^6}={\frac 1 {2^{10}}}(\sqrt5-1)^5$
$16(5\sqrt5-11) =(\sqrt5-1)^5$
Now it is just simple algebra. Notice that raising $\sqrt5$ to an integer power gives either an integer or an integer multiple of $\sqrt5$

Fine, @SammyS, @Frabjous, but, could you give some clue of that algebra? I thoght I had

$$\sqrt[5]\frac{5\sqrt5-11}{2^6}$$

and I had to simplify it, eventually reaching

$$\displaystyle\frac{1}{4}(\sqrt{5}-1)$$

 

[Mark44]

My intention has been to write down the OP's equality's left side as an exponential. And I've also been googling the rules to work out that way.

The point of the other people who replied to your post is that changing the original left side in its radical form to one involving exponents doesn't get you very far in showing that the original equation is an identity.

 

[FactChecker]

Fine, @SammyS, @Frabjous, but, could you give some clue of that algebra? I thoght I had

$$\sqrt[5]\frac{5\sqrt5-11}{2^6}$$

and I had to simplify it, eventually reaching

$$\displaystyle\frac{1}{4}(\sqrt{5}-1)$$

My guess is that this problem was made by raising the RHS to a power, expanding it, and consolidating terms to arrive at the LHS. Then they present it, swapping sides to fool people into thinking that there is a nice method to start from the wrong side and prove it.

 

[FranzS]

Thanks for all the replies.
For context, the expression with the fifth root arises from the calculation of the fifth roots of unity, which I was doing just for fun with no use of trigonometry. After getting that expression, I indeed consulted WA (as @fresh_42 guessed) and it gave me the simplified expression, but I could not see the step-by-step solution (I'm not subscribed).

As I interpret your question, given the radical $R\equiv\sqrt[5]{\frac{5\sqrt{5}-11}{2^{6}}}$, you seek to actually derive the simplified form $R=\frac{1}{4}\left(\sqrt{5}-1\right)$, having no prior knowledge of that expression. That's actually easy to accomplish by making use of the conjugate-element $\overline{R}$ of $R$:$$R=\left(\frac{5\sqrt{5}-11}{2^{6}}\right)^{\frac{1}{5}},\quad\overline{R}=\left(\frac{5\sqrt{5}+11}{2^{6}}\right)^{\frac{1}{5}}\tag{1a,1b}$$from which it follows that $R\overline{R}=\frac{1}{4}$. Then, noting that the simple product $\left(\sqrt{5}-1\right)\left(\sqrt{5}+1\right)=4$, we arrive at the identity:$$R\overline{R}=\frac{1}{4}=\left(\frac{5\sqrt{5}-11}{2^{6}}\right)^{\frac{1}{5}}\left(\frac{5\sqrt{5}+11}{2^{6}}\right)^{\frac{1}{5}}\equiv\left(\frac{\sqrt{5}-1}{4}\right)\left(\frac{\sqrt{5}+1}{4}\right)\tag{2}$$Finally, the most profound observation is that we may individually equate the elements and their conjugates in eq.(2) to develop the two separate identities:$$R=\left(\frac{5\sqrt{5}-11}{2^{6}}\right)^{\frac{1}{5}}\equiv\frac{\sqrt{5}-1}{4},\quad\overline{R}=\left(\frac{5\sqrt{5}+11}{2^{6}}\right)^{\frac{1}{5}}\equiv\frac{\sqrt{5}+1}{4}\tag{3a,3b}$$which both addresses your specific question and yields a free bonus identity!
(Note: I'm no mathematician, but to my understanding, the reason the last factorization step is true stems from Galois theory. Paging @fresh_42?)

Now that's the answer I was looking for, thanks a lot @renormalize !
I guess this approach is pretty systematic (given that conjugation comes as a natural tool to get rid of square roots) except for the step...

[...] Then, noting that the simple product $\left(\sqrt{5}-1\right)\left(\sqrt{5}+1\right)=4$ [...]

... which requires some exeperience and sperimentation. Do you agree?

 

[fresh_42]

... which requires some exeperience and sperimentation. Do you agree?

Whenever you see an expression $a + i b$ where $i$ is a square root of something, you should instinctly multiply it by $a- i b$ regardless what $i$ is.

 

[FranzS]

Whenever you see an expression $a + i b$ where $i$ is a square root of something, you should instinctly multiply it by $a- i b$ regardless what $i$ is.

Agreed. However, the step I was talking about seems to pop up without any systematic approach, right?

 

[renormalize]

Agreed. However, the step I was talking about seems to pop up without any systematic approach, right?

Since you want to find the simplest form of the original radical $R$ involving $\sqrt{5}$, you're looking for an expression $r=a\sqrt{5}+b$ that satisfies $r\overline{r}=R\overline{R}=\frac{1}{4}$. The values $a=\frac{1}{4},b=-\frac{1}{4}$ fill the bill.

 

[fresh_42]

Agreed. However, the step I was talking about seems to pop up without any systematic approach, right?

Yes.

The calculation is post #13 is not really a proof since it uses the result and makes assumptions about conjugate solutions. However, calculating $\left(\dfrac{\sqrt{5}-1}{4}\right)^5$ with the binomial formula is not really hard.

As I said, I tried various methods to solve the problem but didn't find a "trick". The most promising approach is probably proving (with complex numbers)
$$
4\left(\sqrt[5]{\dfrac{5\sqrt{5}-11}{64}}\right)^2+2\left(\sqrt[5]{\dfrac{5\sqrt{5}-11}{64}}\right)-1=0
$$
and then solve $4x^2+2x-1=0.$ I assume that this or something similar was used in the algorithm on WA. But that's a wild guess. I have no idea how their algorithms work.

 

[fresh_42]

Since you want to find the simplest form of the original radical $R$ involving $\sqrt{5}$, you're looking for an expression $r=a\sqrt{5}+b$ that satisfies $r\overline{r}=R\overline{R}=\frac{1}{4}$. The values $a=\frac{1}{4},b=-\frac{1}{4}$ fill the bill.

Yes, I tried. But if you compute $r^5$ you will run into algebraic varieties in $a,b$ that are not really better to solve than the original question.

 

[FranzS]

Since you want to find the simplest form of the original radical $R$ involving $\sqrt{5}$, you're looking for an expression $r=a\sqrt{5}+b$ that satisfies $r\overline{r}=R\overline{R}=\frac{1}{4}$. The values $a=\frac{1}{4},b=-\frac{1}{4}$ fill the bill.

But is it always true that, when $\sqrt{a}$ appears as a nested radicals, it will also appear in the denested expression (if any exists)? Could it not turn in a different radical $\sqrt{b}$?

 

[FactChecker]

I thoght I had

$$\sqrt[5]\frac{5\sqrt5-11}{2^6}$$

and I had to simplify it, eventually reaching

$$\displaystyle\frac{1}{4}(\sqrt{5}-1)$$

I think they calculated it starting from the RHS and then swapped sides to fool you into thinking that. IMHO, that is a cheap trick. There are a lot of things that are easy in one direction but very hard in the other.

 

[fresh_42]

But is it always true that, when $\sqrt{a}$ appears as a nested radicals, it will also appear in the denested expression (if any exists)? Could it not turn in a different radical $\sqrt{b}$?

Yes, $\sqrt[10]{5}$ for example. You cannot know at prior. You want to solve $0=64x^5-5\sqrt{5}+11.$ We know that it has a real root, but we don't know whether it is in $\mathbb{Q}(\sqrt{5}).$

Cardano would have found a way.

 

[renormalize]

The calculation is post #13 is not really a proof since it uses the result and makes assumptions about conjugate solutions.

Can you point out how post #13 "uses the result" and where "assumptions about conjugate solutions" are made? Starting from the original complicated radical element $R$, I simply define a conjugate element $\overline R$, calculate $R\overline{R}=\frac{1}{4}$, postulate a simple expression $r=a\sqrt{5}+b$ that satisfies $r\overline{r}=\frac{1}{4}$, and solve for the values $a=-b=\frac{1}{4}$. (But I readily admit that the final factorization step into separate identities for $R$ and $\overline{R}$, which I only verified numerically, requires advanced mathematics to justify.)

 

[fresh_42]

Can you point out how post #13 "uses the result" and where "assumptions about conjugate solutions" are made? Starting from the original complicated radical element $R$, I simply define a conjugate element $\overline R$, calculate $R\overline{R}=\frac{1}{4}$, postulate a simple expression $r=a\sqrt{5}+b$ that satisfies $r\overline{r}=\frac{1}{4}$, and solve for the values $a=-b=\frac{1}{4}$. (But I readily admit that the final factorization step into separate identities for $R$ and $\overline{R}$, which I only verified numerically, requires advanced mathematics to justify.)

You define $R=\dfrac{1}{4}(\sqrt{5}-1)$ and then operate with $R=\sqrt[5]{\dfrac{5\sqrt{5}-11}{64}}$ and conclude $R\overline{R}=4.$ By that, you also assumed that $\overline{R}$ has the same properties. None of which is known at that stage of the calculation.

 

[renormalize]

You define $R=\dfrac{1}{4}(\sqrt{5}-1)$ and then operate with $R=\sqrt[5]{\dfrac{5\sqrt{5}-11}{64}}$ and conclude $R\overline{R}=4.$ By that, you also assumed that $\overline{R}$ has the same properties. None of which is known at that stage of the calculation.

Sorry, I still don't understand. I define big-$R$ to be $R=\left(\frac{5\sqrt{5}-11}{2^{6}}\right)^{\frac{1}{5}}$ and solve for little-$r$, defined as $r\equiv a\sqrt{5}+b$, to be $\frac{1}{4}\left(\sqrt{5}-1\right)$, and finally verify that $R=r$.

 

[fresh_42]

Sorry, I still don't understand. I define big-$R$ to be $R=\left(\frac{5\sqrt{5}-11}{2^{6}}\right)^{\frac{1}{5}}$ and solve for little-$r$, defined as $r\equiv a\sqrt{5}+b$, to be $\frac{1}{4}\left(\sqrt{5}-1\right)$, and finally verify that $R=r$.

There is no little $r$ on my screen in post #13.

Edit: also $R\overline{R}=r\overline{r}=4$ does not mean $R=r$ and $\overline{R}=\overline{r}$ because $2\cdot 2=4,$ too.

 

[renormalize]

There is no little $r$ on my screen in post #13.

My apologies, I introduced little-$r$ in post #23.
Let me present a revised compilation of my step-by-step reasoning:

1. Define the radical $R$, its conjugate $\overline{R}$, and find their product $R\overline{R}$:$$R\equiv\left(\frac{5\sqrt{5}-11}{2^{6}}\right)^{\frac{1}{5}},\quad\overline{R}\equiv\left(\frac{5\sqrt{5}+11}{2^{6}}\right)^{\frac{1}{5}}\Rightarrow R\overline{R}=\frac{1}{4}\tag{1a,1b,1c}$$
2. Define an expression $r$ and its conjugate $\overline{r}$ such that each is linear in the radical $\sqrt{5}$:$$r\equiv a\sqrt{5}-b,\quad\overline{r}\equiv a\sqrt{5}+b\tag{2a,2b}$$
3. Motivated by eq.(1c), impose the condition:$$r\overline{r}=5a^2-b^2=\frac{1}{4}\tag{3}$$
4. Solve eq.(3) for $b$ and insert it into eq.(2) to find:$$r=a\sqrt{5}-\frac{1}{2}\sqrt{20a^{2}-1}\equiv\frac{\sqrt{5}}{n}-\frac{\sqrt{20-n^{2}}}{2n},\quad\overline{r}=\frac{\sqrt{5}}{n}+\frac{\sqrt{20-n^{2}}}{2n}\tag{4a,4b}$$where $n\equiv a^{-1}$.
5. Now demand that, when $r$ and $\overline{r}$ are raised to integer powers, the resulting expressions should contain no irreducible square-roots other than $\sqrt{5}$. The simplest way to guarantee this is to insist that the coefficients $n^{-1}$ and $(2n)^{-1}\sqrt{20-n^{2}}$ in eqs.(4) themselves contain no irreducible roots whatsoever. Then in particular, $\left|n\right|$ must fall in the range $0<\left|n\right|\leq\sqrt{20}$ so that $\sqrt{20-n^{2}}$ is real and $20-n^{2}$ must be a perfect square. The only two values that satisfy these criteria are $\left|n\right|=2$ and $\left|n\right|=4$, thus yielding the 4 values:$$r_{2}=\frac{\sqrt{5}}{2}-1,\quad\overline{r}_{2}=\frac{\sqrt{5}}{2}+1,\quad r_{4}=\frac{\sqrt{5}-1}{4},\quad\overline{r}_{4}=\frac{\sqrt{5}+1}{4}\tag{5a,5b,5c,5d}$$(Here I've ignored trivial variations involving multiplication by $-1$ on the right sides.)
6. Using Mathematica to raise the right sides of eqs.(5) to the fifth power, I verify the following 4 identities:$$\left(\frac{305\sqrt{5}-682}{2^{5}}\right)^{\frac{1}{5}}\equiv\frac{\sqrt{5}}{2}-1,\quad\left(\frac{305\sqrt{5}+682}{2^{5}}\right)^{\frac{1}{5}}\equiv\frac{\sqrt{5}}{2}+1\tag{6a,6b}$$and$$\left(\frac{5\sqrt{5}-11}{2^{6}}\right)^{\frac{1}{5}}\equiv\frac{\sqrt{5}-1}{4},\quad\left(\frac{5\sqrt{5}+11}{2^{6}}\right)^{\frac{1}{5}}\equiv\frac{\sqrt{5}+1}{4}\tag{6c,6d}$$Under the criteria set-out in step 5, these are the only 4 identities that satisfy $R=r,\overline{R}=\overline{r}, R\overline{R}=r\overline{r}=\frac{1}{4}$. Specifically, eq.(6c) establishes the identity in the OP's first post. QED

As far as I'm aware, I nowhere assume that $r=\frac{\sqrt{5}-1}{4}$. Rather, I derive it from the 3 criteria that: a) $r$ is linear in the radical $\sqrt{5}$, b) $r$ involves no other irreducible radicals, and c) $r\overline{r}=\frac{1}{4}$.

 

[votingmachine]

Proving the identity was never the problem. I'm sure what @FranzS meant was:
$$
0=x^5-\dfrac{5\sqrt{5}-11}{64} \Longrightarrow x=\dfrac{1}{4}\left(\sqrt{5}-1\right) \text{ is one (of 5) solution(s) }
$$
How can we figure out how this root is found?

I tried several methods (your conjugation trick, polynomial division, an ansatz, complex numbers, minimal polynomial ...) but couldn't find one that succeeded. I think meanwhile that @FranzS has taken $\dfrac{\sqrt{5}-1}{4}$ to WA, asked it for $\left(\dfrac{\sqrt{5}-1}{4}\right)^5,$ and posted this question. I finally gave up since I saw no general value in the question: "Does $x^5-\dfrac{5\sqrt{5}-11}{64}\in \mathbb{Q}(\sqrt{5})[x]$ has a root?" when there are technical tools like WA which easily answer that question.

Taking that as:
$$
x^5=\dfrac{5\sqrt{5}-11}{64}
$$

note that the product of the right side by it's conjugate is 2^-10
So:
X^5 times X^5 prime = 2^-10

by the product rule in comment #10, that is the same as:
(X times X prime)^5 = 2^-10

raise both sides to the 5th power:

X times X prime = 2^-2 = 1/4

X and X prime are two of the form:
(n(sqrt(a) + b)) and n((sqrt(a) -b))
and multiplied give:
n^2((a)-b^2)
which equals 1/4

Now it seems intuitive that the a-b^2 part has to be an exponent of 2, as does n. Since both sides seem to require a factorization of 2 ... might be flawed intuition, but ...

So intuitively ...
5-1=4 then times n=1/4 ... which is the solution arrived at.
It seems there are an infinite number of expressions that work though

My apologies for not using the dollars-sign formula stuff. I tried to copy it ... feel free to finish the final step, and to write it legibly.

 

[fresh_42]

My apologies, I introduced little-$r$ in post #23.
Let me present a revised compilation of my step-by-step reasoning:

1. Define the radical $R$, its conjugate $\overline{R}$, and find their product $R\overline{R}$:$$R\equiv\left(\frac{5\sqrt{5}-11}{2^{6}}\right)^{\frac{1}{5}},\quad\overline{R}\equiv\left(\frac{5\sqrt{5}+11}{2^{6}}\right)^{\frac{1}{5}}\Rightarrow R\overline{R}=\frac{1}{4}\tag{1a,1b,1c}$$
2. Define an expression $r$ and its conjugate $\overline{r}$ such that each is linear in the radical $\sqrt{5}$:$$r\equiv a\sqrt{5}-b,\quad\overline{r}\equiv a\sqrt{5}+b\tag{2a,2b}$$
3. Motivated by eq.(1c), impose the condition:$$r\overline{r}=5a^2-b^2=\frac{1}{4}\tag{3}$$
4. Solve eq.(3) for $b$ and insert it into eq.(2) to find:$$r=a\sqrt{5}-\frac{1}{2}\sqrt{20a^{2}-1}\equiv\frac{\sqrt{5}}{n}-\frac{\sqrt{20-n^{2}}}{2n},\quad\overline{r}=\frac{\sqrt{5}}{n}+\frac{\sqrt{20-n^{2}}}{2n}\tag{4a,4b}$$where $n\equiv a^{-1}$.
5. Now demand that, when $r$ and $\overline{r}$ are raised to integer powers, the resulting expressions should contain no irreducible square-roots other than $\sqrt{5}$. The simplest way to guarantee this is to insist that the coefficients $n^{-1}$ and $(2n)^{-1}\sqrt{20-n^{2}}$ in eqs.(4) themselves contain no irreducible roots whatsoever. Then in particular, $\left|n\right|$ must fall in the range $0<\left|n\right|\leq\sqrt{20}$ so that $\sqrt{20-n^{2}}$ is real and $20-n^{2}$ must be a perfect square. The only two values that satisfy these criteria are $\left|n\right|=2$ and $\left|n\right|=4$, thus yielding the 4 values:$$r_{2}=\frac{\sqrt{5}}{2}-1,\quad\overline{r}_{2}=\frac{\sqrt{5}}{2}+1,\quad r_{4}=\frac{\sqrt{5}-1}{4},\quad\overline{r}_{4}=\frac{\sqrt{5}+1}{4}\tag{5a,5b,5c,5d}$$(Here I've ignored trivial variations involving multiplication by $-1$ on the right sides.)
6. Using Mathematica to raise the right sides of eqs.(5) to the fifth power, I verify the following 4 identities:$$\left(\frac{305\sqrt{5}-682}{2^{5}}\right)^{\frac{1}{5}}\equiv\frac{\sqrt{5}}{2}-1,\quad\left(\frac{305\sqrt{5}+682}{2^{5}}\right)^{\frac{1}{5}}\equiv\frac{\sqrt{5}}{2}+1\tag{6a,6b}$$and$$\left(\frac{5\sqrt{5}-11}{2^{6}}\right)^{\frac{1}{5}}\equiv\frac{\sqrt{5}-1}{4},\quad\left(\frac{5\sqrt{5}+11}{2^{6}}\right)^{\frac{1}{5}}\equiv\frac{\sqrt{5}+1}{4}\tag{6c,6d}$$Under the criteria set-out in step 5, these are the only 4 identities that satisfy $R=r,\overline{R}=\overline{r}, R\overline{R}=r\overline{r}=\frac{1}{4}$. Specifically, eq.(6c) establishes the identity in the OP's first post. QED

As far as I'm aware, I nowhere assume that $r=\frac{\sqrt{5}-1}{4}$. Rather, I derive it from the 3 criteria that: a) $r$ is linear in the radical $\sqrt{5}$, b) $r$ involves no other irreducible radicals, and c) $r\overline{r}=\frac{1}{4}$.

This looks ok, but if you use Mathematica to raise the numbers to the fifth power, why not use it to calculate $\left(\dfrac{\sqrt{5}-1}{4}\right)^5$ and find $\dfrac{5\sqrt{5}-11}{2^6}$ or use the binomic formula to do it? I think the detour by the conjugates only makes things more complicated.