Linked Questions

43
votes
5answers
15k views

Strategies to denest nested radicals.

I have recently read some passage about nested radicals, I'm deeply impressed by them. Simple nested radicals $\sqrt{2+\sqrt{2}}$,$\sqrt{3-2\sqrt{2}}$ which the later can be denested into $1-\sqrt{2}$....
10
votes
7answers
41k views

Simplifying $\sqrt{4+2\sqrt{3}}+\sqrt{4-2\sqrt{3}}$

How do I simplify $\sqrt{(4+2\sqrt{3})}+\sqrt{(4-2\sqrt{3})}$? I've tried to make it $x$ and square both sides but I got something extremely complicated and it didn't look right. I got $2\sqrt{3}$ ...
13
votes
5answers
11k views

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

$$\sqrt[4]{161-72 \sqrt{5}}$$ 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 ...
5
votes
4answers
13k views

Proving that the number $\sqrt[3]{7 + \sqrt{50}} + \sqrt[3]{7 - 5\sqrt{2}}$ is rational

I've been struggling to show that the number $\sqrt[3]{7 + \sqrt{50}} + \sqrt[3]{7 - 5\sqrt{2}}$ is rational. I would like to restructure it to prove it, but I can't find anything besides $\sqrt{50} =...
4
votes
5answers
2k views

Simple solving Skanavi book exercise: $\sqrt[3]{9+\sqrt{80}}+\sqrt[3]{9-\sqrt{80}}$ [closed]

Simple way to solve this exercise $$ x = \sqrt[3]{9+\sqrt{80}}+\sqrt[3]{9-\sqrt{80}} $$
4
votes
6answers
2k views

Show that $\sqrt[3]{3\sqrt{21} + 8} - \sqrt[3]{3\sqrt{21} - 8} = 1$

Show that $$\sqrt[3]{3\sqrt{21} + 8} - \sqrt[3]{3\sqrt{21} - 8} = 1$$ Playing around with the expression, I found a proof which I will post as an answer. I'm asking this question because I would ...
7
votes
2answers
3k views

Denesting a square root: $\sqrt{7 + \sqrt{14}}$

Write: $$\sqrt{7 + \sqrt{14}} = a + b\sqrt{c}$$ Form. $$7 + \sqrt{14} = a^2 + 2ab\sqrt{c} + b^2c$$ $a^2 + b^2c = 7$ and $2ab = 1$, and $c = 14$ But that doesn't seem right as $a, b,$ wont be ...
4
votes
6answers
429 views

How do I show that $\sqrt{5+\sqrt{24}} = \sqrt{3}+\sqrt{2}$

According to wolfram alpha this is true: $\sqrt{5+\sqrt{24}} = \sqrt{3}+\sqrt{2}$ But how do you show this? I know of no rules that works with addition inside square roots. I noticed I could do this:...
4
votes
4answers
177 views

Calculate simple expression: $\sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}}$ [duplicate]

Tell me please, how calculate this expression: $$ \sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}} $$ The result should be a number. I try this: $$ \frac{\left(\sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - ...
1
vote
3answers
174 views

Expressing $\sqrt[3]{7+5\sqrt{2}}$ in the form $x+y\sqrt{2}$ [closed]

Express $\sqrt[3]{(7+5\sqrt{2})}$ in the form $x+y\sqrt{2}$ with $x$ and $y$ rational numbers. I.e. Show that it is $1+\sqrt{2}$.
4
votes
3answers
221 views

simplify $\sqrt[3]{5+2\sqrt{13}}+\sqrt[3]{5-2\sqrt{13}}$ [closed]

simplify $$\sqrt[3]{5+2\sqrt{13}}+\sqrt[3]{5-2\sqrt{13}}$$ 1.$\frac{3}{2}$ 2.$\frac{\sqrt[3]{65}}{4}$ 3.$\sqrt[3]{2}$ 4.$1$ I equal it to $\sqrt[3]{a}+\sqrt[3]{b}$ but I cant find $...
3
votes
1answer
373 views

Rewritting a messy cubic root

Is there a really quick way of showing that: $$\sqrt[3]{49-25\sqrt{2}}$$ Can be written in the form: $$a+b\sqrt{2}$$ Is there a way to generalize which integers $a$ and $b$ can be rewritten in ...
2
votes
2answers
345 views

Convert from Nested Square Roots to Sum of Square Roots

I am looking for a way to easily discover how to go from a nested root to a sum of roots. For example, $$\sqrt{10-2\sqrt{21}}=\sqrt{3}-\sqrt{7}$$ I know that if I set $\alpha=\sqrt{10-2\sqrt{21}}$,...
2
votes
3answers
140 views

Express this sum of radicals as an integer? [duplicate]

I have read somewhere that the radical $\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}=1$ and I don't understand it. How do you solve this(when the RHS is unknown)?
3
votes
2answers
114 views

When is $\sqrt[3]{a+\sqrt b}+\sqrt[3]{a-\sqrt b}$ an integer? [duplicate]

I saw a Youtube video in which it was shown that $$(7+50^{1/2})^{1/3}+(7-50^{1/2})^{1/3}=2$$ Since there are multiple values we can choose for the $3$rd root of a number, it would also make more sense ...

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