Definition of the exponential function, Zorich - 1
$begingroup$
The image referenced is from "Mathematical Analysis - I, Zorich" pages 118-119. It provides the context to the doubts below:
Doubts:
1) How is $$text{Indeed } a^{(mk)/(nk)}>0,kin Bbb Z ldots$$ true when, so far, only forms $a^{m/n}, m in Bbb Z, n in Bbb N$ have been defined? He has an integer in the denominator of the exponent when he's only defined a form with natural numbers in the denominator.
2)How is $$(a^{(mk)/(nk)})^{nk}=Bigl((a^{1/(nk)})^{mk}Bigr)^{nk}$$ when, again, it has only been established so far that $a^{m/n}=(a^{1/n})^m text{ for }min Bbb Z,n in Bbb N$ ? I dont see how the $mk$ could have been brought out of the brackets.
Reference Material:
real-analysis
asked Jan 24 at 4:11
Ashwin BAshwin B
597
$endgroup$
add a comment |
$begingroup$
The image referenced is from "Mathematical Analysis - I, Zorich" pages 118-119. It provides the context to the doubts below:
Doubts:
1) How is $$text{Indeed } a^{(mk)/(nk)}>0,kin Bbb Z ldots$$ true when, so far, only forms $a^{m/n}, m in Bbb Z, n in Bbb N$ have been defined? He has an integer in the denominator of the exponent when he's only defined a form with natural numbers in the denominator.
2)How is $$(a^{(mk)/(nk)})^{nk}=Bigl((a^{1/(nk)})^{mk}Bigr)^{nk}$$ when, again, it has only been established so far that $a^{m/n}=(a^{1/n})^m text{ for }min Bbb Z,n in Bbb N$ ? I dont see how the $mk$ could have been brought out of the brackets.
Reference Material:
real-analysis
asked Jan 24 at 4:11
Ashwin BAshwin B
597
$endgroup$
add a comment |
$begingroup$
The image referenced is from "Mathematical Analysis - I, Zorich" pages 118-119. It provides the context to the doubts below:
Doubts:
1) How is $$text{Indeed } a^{(mk)/(nk)}>0,kin Bbb Z ldots$$ true when, so far, only forms $a^{m/n}, m in Bbb Z, n in Bbb N$ have been defined? He has an integer in the denominator of the exponent when he's only defined a form with natural numbers in the denominator.
2)How is $$(a^{(mk)/(nk)})^{nk}=Bigl((a^{1/(nk)})^{mk}Bigr)^{nk}$$ when, again, it has only been established so far that $a^{m/n}=(a^{1/n})^m text{ for }min Bbb Z,n in Bbb N$ ? I dont see how the $mk$ could have been brought out of the brackets.
Reference Material:
real-analysis
asked Jan 24 at 4:11
Ashwin BAshwin B
597
$endgroup$
The image referenced is from "Mathematical Analysis - I, Zorich" pages 118-119. It provides the context to the doubts below:
Doubts:
1) How is $$text{Indeed } a^{(mk)/(nk)}>0,kin Bbb Z ldots$$ true when, so far, only forms $a^{m/n}, m in Bbb Z, n in Bbb N$ have been defined? He has an integer in the denominator of the exponent when he's only defined a form with natural numbers in the denominator.
2)How is $$(a^{(mk)/(nk)})^{nk}=Bigl((a^{1/(nk)})^{mk}Bigr)^{nk}$$ when, again, it has only been established so far that $a^{m/n}=(a^{1/n})^m text{ for }min Bbb Z,n in Bbb N$ ? I dont see how the $mk$ could have been brought out of the brackets.
Reference Material:
real-analysis
real-analysis
asked Jan 24 at 4:11
Ashwin BAshwin B
597
asked Jan 24 at 4:11
Ashwin BAshwin B
597
asked Jan 24 at 4:11
Ashwin BAshwin B
597
asked Jan 24 at 4:11
Ashwin BAshwin B
597
asked Jan 24 at 4:11
Ashwin BAshwin B
597
597
add a comment |
add a comment |
0
active
oldest
votes
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%2f3085449%2fdefinition-of-the-exponential-function-zorich-1%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
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%2f3085449%2fdefinition-of-the-exponential-function-zorich-1%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
