Mixing
Is it guaranteed that $sqrt[3]{a+sqrt{b}}$ can be denested with or without complex numbers?
$begingroup$
I tried to use the cubic formula before, but was always stuck at simplifying the cube root. I had learned that you can always simplify $sqrt{a+sqrt{b}}$ by solving, but can I always simplify $sqrt[3]{a+sqrt{b}}$, too, or is there a restriction?
Edit:
It can contain complex numbers, and I mean denesting by simplifying. $sqrt{a+sqrt{b}}$ is the same as solving m+n=a, and $mn={bover4}$.
Second Edit:
An example will be denesting $sqrt{1+sqrt{2}}$. $sqrt{1+sqrt{2}}$ will be $sqrt{m}+sqrt{n}$, where m+n=1, and $mn={1over2}$. Squaring the first equation then subtracting the second equation 4 times, m-n=$i$. So, $sqrt{1+sqrt{2}}$=$sqrt{1+iover2}+sqrt{1-iover2}$.
radicals nested-radicals
asked Jan 31 at 0:31
Math LoverMath Lover
17410
$endgroup$
add a comment |
$begingroup$
I tried to use the cubic formula before, but was always stuck at simplifying the cube root. I had learned that you can always simplify $sqrt{a+sqrt{b}}$ by solving, but can I always simplify $sqrt[3]{a+sqrt{b}}$, too, or is there a restriction?
Edit:
It can contain complex numbers, and I mean denesting by simplifying. $sqrt{a+sqrt{b}}$ is the same as solving m+n=a, and $mn={bover4}$.
Second Edit:
An example will be denesting $sqrt{1+sqrt{2}}$. $sqrt{1+sqrt{2}}$ will be $sqrt{m}+sqrt{n}$, where m+n=1, and $mn={1over2}$. Squaring the first equation then subtracting the second equation 4 times, m-n=$i$. So, $sqrt{1+sqrt{2}}$=$sqrt{1+iover2}+sqrt{1-iover2}$.
radicals nested-radicals
asked Jan 31 at 0:31
Math LoverMath Lover
17410
$endgroup$
$begingroup$
I feel a bit confused about what you mean by "complex numbers". So can you please show perhaps an example of an acceptable denesting of $sqrt{1+sqrt{2}}$?
$endgroup$
– Edward H.
Feb 1 at 1:28
add a comment |
$begingroup$
I tried to use the cubic formula before, but was always stuck at simplifying the cube root. I had learned that you can always simplify $sqrt{a+sqrt{b}}$ by solving, but can I always simplify $sqrt[3]{a+sqrt{b}}$, too, or is there a restriction?
Edit:
It can contain complex numbers, and I mean denesting by simplifying. $sqrt{a+sqrt{b}}$ is the same as solving m+n=a, and $mn={bover4}$.
Second Edit:
An example will be denesting $sqrt{1+sqrt{2}}$. $sqrt{1+sqrt{2}}$ will be $sqrt{m}+sqrt{n}$, where m+n=1, and $mn={1over2}$. Squaring the first equation then subtracting the second equation 4 times, m-n=$i$. So, $sqrt{1+sqrt{2}}$=$sqrt{1+iover2}+sqrt{1-iover2}$.
radicals nested-radicals
asked Jan 31 at 0:31
Math LoverMath Lover
17410
$endgroup$
I tried to use the cubic formula before, but was always stuck at simplifying the cube root. I had learned that you can always simplify $sqrt{a+sqrt{b}}$ by solving, but can I always simplify $sqrt[3]{a+sqrt{b}}$, too, or is there a restriction?
Edit:
It can contain complex numbers, and I mean denesting by simplifying. $sqrt{a+sqrt{b}}$ is the same as solving m+n=a, and $mn={bover4}$.
Second Edit:
An example will be denesting $sqrt{1+sqrt{2}}$. $sqrt{1+sqrt{2}}$ will be $sqrt{m}+sqrt{n}$, where m+n=1, and $mn={1over2}$. Squaring the first equation then subtracting the second equation 4 times, m-n=$i$. So, $sqrt{1+sqrt{2}}$=$sqrt{1+iover2}+sqrt{1-iover2}$.
radicals nested-radicals
radicals nested-radicals
asked Jan 31 at 0:31
Math LoverMath Lover
17410
asked Jan 31 at 0:31
Math LoverMath Lover
17410
edited Feb 1 at 19:40
Math Lover
asked Jan 31 at 0:31
Math LoverMath Lover
17410
asked Jan 31 at 0:31
Math LoverMath Lover
17410
asked Jan 31 at 0:31
Math LoverMath Lover
17410
17410
$begingroup$
I feel a bit confused about what you mean by "complex numbers". So can you please show perhaps an example of an acceptable denesting of $sqrt{1+sqrt{2}}$?
$endgroup$
– Edward H.
Feb 1 at 1:28
add a comment |
$begingroup$
I feel a bit confused about what you mean by "complex numbers". So can you please show perhaps an example of an acceptable denesting of $sqrt{1+sqrt{2}}$?
$endgroup$
– Edward H.
Feb 1 at 1:28
$begingroup$
I feel a bit confused about what you mean by "complex numbers". So can you please show perhaps an example of an acceptable denesting of $sqrt{1+sqrt{2}}$?
$endgroup$
– Edward H.
Feb 1 at 1:28
$begingroup$
I feel a bit confused about what you mean by "complex numbers". So can you please show perhaps an example of an acceptable denesting of $sqrt{1+sqrt{2}}$?
$endgroup$
– Edward H.
Feb 1 at 1:28
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
If you mean "can be simplified" as deleting of nested radicals then it's not always possible.
For example, for $sqrt{3-2sqrt2}$ we can do it, but for $sqrt{sqrt2-1}$ it's impossible.
For $sqrt[3]{5sqrt2-7}$ we can do it, but for $sqrt[3]{sqrt2-1}$ it's impossible.
For $sqrt[3]{sqrt[3]2-1}$ we can do it, but for $sqrt[3]{sqrt[3]2+1}$ it's impossible.
answered Jan 31 at 1:03
Michael RozenbergMichael Rozenberg
110k1896201
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3094325%2fis-it-guaranteed-that-sqrt3a-sqrtb-can-be-denested-with-or-without-com%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
If you mean "can be simplified" as deleting of nested radicals then it's not always possible.
For example, for $sqrt{3-2sqrt2}$ we can do it, but for $sqrt{sqrt2-1}$ it's impossible.
For $sqrt[3]{5sqrt2-7}$ we can do it, but for $sqrt[3]{sqrt2-1}$ it's impossible.
For $sqrt[3]{sqrt[3]2-1}$ we can do it, but for $sqrt[3]{sqrt[3]2+1}$ it's impossible.
answered Jan 31 at 1:03
Michael RozenbergMichael Rozenberg
110k1896201
$endgroup$
add a comment |
$begingroup$
If you mean "can be simplified" as deleting of nested radicals then it's not always possible.
For example, for $sqrt{3-2sqrt2}$ we can do it, but for $sqrt{sqrt2-1}$ it's impossible.
For $sqrt[3]{5sqrt2-7}$ we can do it, but for $sqrt[3]{sqrt2-1}$ it's impossible.
For $sqrt[3]{sqrt[3]2-1}$ we can do it, but for $sqrt[3]{sqrt[3]2+1}$ it's impossible.
answered Jan 31 at 1:03
Michael RozenbergMichael Rozenberg
110k1896201
$endgroup$
add a comment |
$begingroup$
If you mean "can be simplified" as deleting of nested radicals then it's not always possible.
For example, for $sqrt{3-2sqrt2}$ we can do it, but for $sqrt{sqrt2-1}$ it's impossible.
For $sqrt[3]{5sqrt2-7}$ we can do it, but for $sqrt[3]{sqrt2-1}$ it's impossible.
For $sqrt[3]{sqrt[3]2-1}$ we can do it, but for $sqrt[3]{sqrt[3]2+1}$ it's impossible.
answered Jan 31 at 1:03
Michael RozenbergMichael Rozenberg
110k1896201
$endgroup$
If you mean "can be simplified" as deleting of nested radicals then it's not always possible.
For example, for $sqrt{3-2sqrt2}$ we can do it, but for $sqrt{sqrt2-1}$ it's impossible.
For $sqrt[3]{5sqrt2-7}$ we can do it, but for $sqrt[3]{sqrt2-1}$ it's impossible.
For $sqrt[3]{sqrt[3]2-1}$ we can do it, but for $sqrt[3]{sqrt[3]2+1}$ it's impossible.
answered Jan 31 at 1:03
Michael RozenbergMichael Rozenberg
110k1896201
answered Jan 31 at 1:03
Michael RozenbergMichael Rozenberg
110k1896201
answered Jan 31 at 1:03
Michael RozenbergMichael Rozenberg
110k1896201
answered Jan 31 at 1:03
Michael RozenbergMichael Rozenberg
110k1896201
110k1896201
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3094325%2fis-it-guaranteed-that-sqrt3a-sqrtb-can-be-denested-with-or-without-com%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
I feel a bit confused about what you mean by "complex numbers". So can you please show perhaps an example of an acceptable denesting of $sqrt{1+sqrt{2}}$?
$endgroup$
– Edward H.
Feb 1 at 1:28