MHB How can we solve a radical equation without using a calculator?

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The discussion focuses on solving the radical equation without a calculator by demonstrating that the right-hand side (RHS) equals the left-hand side (LHS) through algebraic manipulation. It shows that the expression \(2 \pm \sqrt{5}\) can be represented as \(\left(\frac{1 \pm \sqrt{5}}{2}\right)^3\). The conversation then delves into the tedious process of taking the cube root of both sides and solving for variables \(a\) and \(b\) in the context of quadratic integers. By applying the norm of quadratic integers, the participants derive a cubic equation to find rational roots, ultimately concluding that \(\sqrt[3]{2 \pm \sqrt{5}} = \frac{1}{2} \pm \frac{1}{2} \sqrt{5}\). The overall sentiment acknowledges the complexity of the algebra involved in this solution.
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Show that the RHS = LHS without using a calculator.

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I would observe that:

$$\left(1\pm\sqrt{5}\right)^3=1\pm3\sqrt{5}+3\cdot5\pm5\sqrt{5}=16\pm8\sqrt{5}=8\left(2\pm\sqrt{5}\right)$$

Hence:

$$2\pm\sqrt{5}=\left(\frac{1\pm\sqrt{5}}{2}\right)^3$$
 
MarkFL said:
I would observe that:

$$\left(1\pm\sqrt{5}\right)^3=1\pm3\sqrt{5}+3\cdot5\pm5\sqrt{5}=16\pm8\sqrt{5}=8\left(2\pm\sqrt{5}\right)$$

Hence:

$$2\pm\sqrt{5}=\left(\frac{1\pm\sqrt{5}}{2}\right)^3$$

What would the algebra look like if I take the cube root on both sides? Is it tedious?
 
Last edited:
That's not necessary. Put $\left(\frac{1\pm\sqrt5}{2}\right)^3$ instead of $2\pm\sqrt5$ in $\sqrt[3]{2+\sqrt5}+\sqrt[3]{2-\sqrt5}=1$. What do you get?

If you mean taking the cube root of both sides of $2\pm\sqrt{5}=\left(\frac{1\pm\sqrt{5}}{2}\right)^3$, you'd arrive at the solution.
 
RTCNTC said:
What would the algebra look like if I take the cube root on both sides? Is it tedious?

Let's assume that the cube root is in the same Ring of Quadratic Integers.
That is, we can write the cube root $\sqrt[3]{2\pm \sqrt 5}$ as $a+b\sqrt 5$.

Then to take the cube root, we can solve a and b from:
$$(a+b\sqrt 5)^3 = 2 \pm \sqrt 5$$
We have:
$$(a+b\cdot\sqrt 5)^3 = a^3+3a^2b\cdot\sqrt 5 + 3ab^2\cdot 5 + b^2\cdot 5\sqrt 5 = a(a^2+15b^2) + b(3a^2 + 5b^2)\sqrt 5$$
So:
$$\begin{cases}a(a^2+15b^2) = 2 \\ b(3a^2 + 5b^2) = \pm 1
\end{cases} \tag 1$$
Unfortunately this is quite tedious to solve. (Worried)

Luckily we can draw on the theory of Quadratic Integers to make it a bit easier.
That's by using the norm $N$, which is defined as $N(x+y\sqrt 5)=x^2-5y^2$.
The norm $N(a+b\sqrt 5)$, when raised to the power of 3, must then be equal to $N(2\pm \sqrt 5)$.
Thus, we get the additional identity:
$$N(a+b\sqrt 5)^3 = N(2\pm \sqrt 5) \quad\Rightarrow\quad (a^2 - 5b^2)^3 = 2^2 - 5(\pm 1)^2 = -1$$
This implies:
$$a^2-5b^2 = -1 \quad\Rightarrow\quad 5b^2 = a^2+1 \tag 2$$
Substituting in (1) yields:
$$a(a^2+3\cdot 5b^2)= a(a^2+3(a^2+1))=4a^3+3a=2 \quad\Rightarrow\quad 4a^3+3a-2=0$$

Next, from the Rational Root Theorem we know that any rational root must be one of:
$$\pm 2, \pm 1, \pm\frac 12, \pm\frac 14$$
And indeed, $a=\frac 12$ fits.
Substituting back into (2) yields:
$$5b^2 = \Big(\frac 12\Big)^2+1 = \frac 54 \quad\Rightarrow\quad b=\pm\frac 12$$

So we find that:
$$\sqrt[3]{2\pm \sqrt 5} = \frac 12 \pm \frac 12 \sqrt 5 \tag 3$$TL;DR: Yep, it's tedious to take the cube root. (Angel)
 
I like Serena said:
Let's assume that the cube root is in the same Ring of Quadratic Integers.
That is, we can write the cube root $\sqrt[3]{2\pm \sqrt 5}$ as $a+b\sqrt 5$.

Then to take the cube root, we can solve a and b from:
$$(a+b\sqrt 5)^3 = 2 \pm \sqrt 5$$
We have:
$$(a+b\cdot\sqrt 5)^3 = a^3+3a^2b\cdot\sqrt 5 + 3ab^2\cdot 5 + b^2\cdot 5\sqrt 5 = a(a^2+15b^2) + b(3a^2 + 5b^2)\sqrt 5$$
So:
$$\begin{cases}a(a^2+15b^2) = 2 \\ b(3a^2 + 5b^2) = \pm 1
\end{cases} \tag 1$$
Unfortunately this is quite tedious to solve. (Worried)

Luckily we can draw on the theory of Quadratic Integers to make it a bit easier.
That's by using the norm $N$, which is defined as $N(x+y\sqrt 5)=x^2-5y^2$.
The norm $N(a+b\sqrt 5)$, when raised to the power of 3, must then be equal to $N(2\pm \sqrt 5)$.
Thus, we get the additional identity:
$$N(a+b\sqrt 5)^3 = N(2\pm \sqrt 5) \quad\Rightarrow\quad (a^2 - 5b^2)^3 = 2^2 - 5(\pm 1)^2 = -1$$
This implies:
$$a^2-5b^2 = -1 \quad\Rightarrow\quad 5b^2 = a^2+1 \tag 2$$
Substituting in (1) yields:
$$a(a^2+3\cdot 5b^2)= a(a^2+3(a^2+1))=4a^3+3a=2 \quad\Rightarrow\quad 4a^3+3a-2=0$$

Next, from the Rational Root Theorem we know that any rational root must be one of:
$$\pm 2, \pm 1, \pm\frac 12, \pm\frac 14$$
And indeed, $a=\frac 12$ fits.
Substituting back into (2) yields:
$$5b^2 = \Big(\frac 12\Big)^2+1 = \frac 54 \quad\Rightarrow\quad b=\pm\frac 12$$

So we find that:
$$\sqrt[3]{2\pm \sqrt 5} = \frac 12 \pm \frac 12 \sqrt 5 \tag 3$$TL;DR: Yep, it's tedious to take the cube root. (Angel)

Wonderfully informative reply. Thanks.
 
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