Potentiae exponentis rationalis et radicalia

Svmmarivm

Haec lectio potentias ad exponentes rationales per radicem principalem extendit, et demonstrat definitionem non pendere a fractione qva exponens scribitvr, legesqve potentiarvm exponentis integri valere pergere. Deinde leges radicalivm cvm svis hypothesibvs de signo statvit, errorem radicis in svmmam distribvendae describit, artesqve calcvlandi cvm radicalibvs docet: factores extrahere, denominatores rationales reddere per conivgata et per differentiam cvborvm, nvmeros sine machinvla calcvlatoria comparare, radicalia dvplicia simplificare.

Proposita discendi

Potentiam exponentis rationalis basis positivae definire, ostendere eam a fractione electa non pendere, eivsqve leges adhibere.
Leges radicalivm cvm svis hypothesibvs de signo envntiare, praesertim $\sqrt[n]{a^n} = \abs{a}$ pro indice pari, et errores vsitatos agnoscere.
Radicalia simplificare et denominatores rationales reddere, qvi monomia, binomia avt radices cvbicas continentes svnt.
Nvmeros cvm radicalibvs comparare eos ad potentias aptas elevando, et radicalia dvplicia simplificare, cvm fieri potest.

Potentiae exponentis rationalis

In lectione de generibvs nvmerorvm definitae svnt potentiae exponentis integri et radix principalis: pro natvrali $n \geq 2$ et reali $b \geq 0$ , nvmervs $\sqrt[n]{b}$ est vnicvs nvmervs realis non negativvs cvivs potentia $n$ -ta est $b$ . Ibi demonstratae svnt, pro basibvs rationalibvs, leges potentiarvm exponentis integri; demonstrationes earvm solis proprietatibvs additionis et mvltiplicationis vtvntvr, ita vt pariter pro basibvs realibvs valeant. Ibi etiam demonstrata est vnicitas radicis principalis, a qva omnia qvae seqvvntvr pendent, eivsqve exsistentia admissa, qvae ex completvdine nvmerorvm realivm seqvitvr et in lectione de recta reali demonstrata est.

Potentiae et radices operationes inversae svnt, et eas vna notatione conivngere convenit. Si lex potentiae potentiae etiam pro exponentibvs fractis valere debet, nvmervs $a^{\frac{1}{q}}$ condicioni $\left(a^{\frac{1}{q}}\right)^q = a^1 = a$ satisfacere debet; id est, radix $q$ -ta nvmeri $a$ esse debet, et natvralis electio est radix principalis. Eodem modo $a^{\frac{p}{q}}$ esse debet potentia $p$ -ta nvmeri $a^{\frac{1}{q}}$ , vel radix $q$ -ta nvmeri $a^p$ . Haec observatio nihil demonstrat, qvia leges qvae obtinendae svnt svpponit; definitionem tantvm svggerit.

Definitio (potentia exponentis rationalis). Sit $a > 0$ nvmervs realis, $p$ integer et $q$ natvralis. Potentia exponentis rationalis $a^{\frac{p}{q}}$ est radix $q$ -ta principalis nvmeri $a^p$ . Pro $q = 1$ intellegitvr radicem indicis $1$ nvmeri esse ipsvm nvmervm, ita vt definitio cvm definitione potentiae exponentis integri congrvat.

a^{\frac{p}{q}} = \sqrt[q]{a^p}

Radix exsistit qvia $a^p$ positivvs est, et eadem de cavsa positiva est: si nvlla esset, etiam potentia eivs $q$ -ta nvlla esset. Exempli gratia, $8^{\frac{2}{3}} = \sqrt[3]{64} = 4$ et $27^{-\frac{1}{3}} = \sqrt[3]{\frac{1}{27}} = \frac{1}{3}$ . In primo exemplo idem eventvs obtinetvr si privs radix extrahitvr et postea ad potentiam elevatvr, qvia $\left(\sqrt[3]{8}\right)^2 = 2^2 = 4$ ; ordo dvarvm operationvm nihil refert.

LemmaRadix potentiae

Pro omni reali $a > 0$ , omni integro $p$ et omni natvrali $q$ , radix $q$ -ta nvmeri $a^p$ est potentia $p$ -ta radicis $q$ -tae nvmeri $a$ .

\sqrt[q]{a^p} = \left(\sqrt[q]{a}\right)^p

Confirmatvr membrvm altervm signvm et potentiam habere qvae primvm definivnt, et vnicitas radicis principalis adhibetvr. Figvra ratiocinationem comitatvr cvm $a = 2$ , $q = 2$ et $p = 3$ : cvrva $y = t^q$ in positivis crescit et qvamqve rectam horizontalem altitvdinis positivae semel tantvm secat.

Demonstratio

$r = \sqrt[q]{a}, \quad r > 0, \quad \dato{rq}{r^q = a}$
Si $q = 1$ , ambo membra svnt $a^p$ . Si $q \geq 2$ , sit $r$ radix $q$ -ta principalis nvmeri $a$ : positiva est, qvia si nvlla esset, etiam $a$ nvllvs esset, et potentia eivs $q$ -ta est $a$ .
$\dato{pos}{r^p > 0}$
Nvmervs $r^p$ positivvs est, qvia omnis potentia exponentis integri nvmeri positivi positiva est: si $p$ natvralis est, per axioma prodvcti; si $p = 0$ , valet $1$ ; et si $p$ negativvs est, est inversvm nvmeri positivi.
$\left(r^p\right)^q = r^{pq} = \left(\resaltar{r^q}\right)^p = \dato{ap}{\resaltar{a^p}}$
Per legem potentiae potentiae exponentis integri, vtroqve ordine adhibitam, potentia $q$ -ta nvmeri $r^p$ est $a^p$ .
$\sqrt[q]{a^p} = \resaltar{r^p} = \left(\sqrt[q]{a}\right)^p$
Ita $r^p$ est nvmervs positivvs cvivs potentia $q$ -ta est $a^p$ . Per vnicitatem radicis principalis, est radix $q$ -ta nvmeri $a^p$ .

Definitio a fractione non pendet

Idem nvmervs rationalis mvltis fractionibvs scribitvr: $\frac{1}{2} = \frac{2}{4} = \frac{3}{6}$ . Definitio svperior certa fractione formatvr, et nihil antea praestat dvas scriptvras eivsdem exponentis evndem nvmervm dare; exempli gratia, $2^{\frac{1}{2}} = \sqrt{2}$ , cvm tamen $2^{\frac{2}{4}} = \sqrt[4]{4}$ . Si nvmeros diversos darent, symbolvm $a^x$ pro rationali $x$ sensvm non haberet, sed tantvm pro singvlis eivs fractionibvs.

TheoremaIndependentia a fractione

Sit $a > 0$ . Si $p$ et $r$ integri svnt, $q$ et $s$ natvrales svnt et $\frac{p}{q} = \frac{r}{s}$ , tvm $\sqrt[q]{a^p} = \sqrt[s]{a^r}$ . Consilivm est dvos nvmeros ad potentiam $qs$ elevare, qvae ambas radices tollit, et vnicitatem radicis principalis adhibere. Figvra ratiocinationem comitatvr cvm $a = 2$ , $\frac{p}{q} = \frac{1}{2}$ et $\frac{r}{s} = \frac{2}{4}$ : qvaeqve potentia cvrva crescens est, qvae qvamqve rectam horizontalem semel tantvm secat.

Demonstratio

$u = \sqrt[q]{a^p}, \quad v = \sqrt[s]{a^r}, \quad \dato{du}{u^q = a^p}, \quad \dato{dv}{v^s = a^r}$
Appellamvs $u$ et $v$ nvmeros qvos dvae scriptvrae dant. Ambo positivi svnt et, per definitionem radicis, aeqvalitatibvs seqventibvs satisfacivnt.
$u^{qs} = \left(\resaltar{u^q}\right)^s = \left(a^p\right)^s = \dato{pu}{\resaltar{a^{ps}}}$
Elevamvs $u$ ad potentiam $qs$ per legem potentiae potentiae, et pro $u^q$ valorem eivs ponimvs.
$v^{qs} = \left(\resaltar{v^s}\right)^q = \left(a^r\right)^q = \dato{pv}{\resaltar{a^{rq}}}$
Eodem modo cvm $v$ , privs exponentem $s$ conivngendo.
$\frac{p}{q} = \frac{r}{s} \leftrightarrow \dato{cr}{\resaltar{ps = qr}}$
Fractiones aeqvales svnt, qvod significat prodvcta earvm decvssata congrvere.
$u^{qs} = a^{\resaltar{ps}} = a^{\resaltar{qr}} = v^{qs}$
Vnde $u$ et $v$ eandem potentiam indicis $qs$ habent.
$\sqrt[q]{a^p} = \sqrt[s]{a^r}$
Si $qs = 1$ , hoc dicit $u = v$ . Si $qs \geq 2$ , $u$ et $v$ svnt dvo nvmeri positivi eandem potentiam exponentis $qs$ habentes, et per vnicitatem radicis principalis aeqvales svnt.

Ergo pro omni rationali $x$ nvmervs $a^x$ bene definitvs est: $x$ qvolibet modo vt fractio scribitvr, et eventvs semper idem est. Praesertim omnis exponens rationalis cvm denominatore natvrali scribi potest, et dvo exponentes cvm denominatore commvni scribi possvnt, qvod in demonstrationibvs seqventibvs adhibetvr.

Hypothesis $a > 0$ non est libido. Basi negativa, radix indicis imparis exsistit, sed eventvs a fractione penderet: $\sqrt[3]{-8} = -2$ , cvm tamen $\sqrt[6]{(-8)^2} = \sqrt[6]{64} = 2$ , qvamvis $\frac{1}{3} = \frac{2}{6}$ . Qvam ob rem potentiae exponentis rationalis cvm basi negativa non definivntvr, et pro nvmeris negativis radix scribitvr, vt $\sqrt[3]{-8}$ . Basi nvlla definitvr $0^x = 0$ pro omni rationali $x > 0$ , et exponentes negativi vel nvlli non definivntvr.

Leges exponentivm rationalivm

Definitione firmata, confirmatvr leges potentiarvm exponentis integri valere pergere.

TheoremaLeges exponentivm rationalivm

Pro omnibvs realibvs $a > 0$ et $b > 0$ et omnibvs rationalibvs $x$ et $y$ valent $a^x \cdot a^y = a^{x + y}$ , $\left(a^x\right)^y = a^{xy}$ et $(ab)^x = a^x \cdot b^x$ . Itaqve etiam $a^{-x} = \frac{1}{a^x}$ et $\frac{a^x}{a^y} = a^{x - y}$ . Dvae priores leges demonstrantvr; tertia ex lege prodvcti radicalivm seqvitvr, in capite seqventi. Figvra valores $2^{\frac{k}{6}}$ ostendit: qvisqve est praecedens per $\sqrt[6]{2}$ mvltiplicatvs, et exponentes addvntvr sicvt in scalis ascenditvr.

Demonstratio

$x = \frac{p}{q}, \quad y = \frac{r}{q}, \quad u = \sqrt[q]{a}, \quad \dato{lx}{a^x = u^p}, \quad \dato{ly}{a^y = u^r}$
Prodvctvm. Scribimvs $x$ et $y$ cvm denominatore commvni $q$ , qvod per theorema praecedens licet, et appellamvs $u$ radicem $q$ -tam nvmeri $a$ . Per lemma de radice potentiae, ambae potentiae svnt potentiae integrae nvmeri $u$ .
$a^x \cdot a^y = \resaltar{u^p \cdot u^r} = u^{p + r} = \resaltar{a^{\frac{p + r}{q}}} = a^{x + y}$
Per legem prodvcti exponentis integri, exponentes addvntvr; et rvrsvs per lemma, eventvs est $a^{x + y}$ , qvia $x + y = \frac{p + r}{q}$ .
$\left(a^x\right)^y = \sqrt[s]{\left(\resaltar{u^p}\right)^r} = \dato{w}{\resaltar{\sqrt[s]{u^{pr}}}} = w$
Potentia potentiae. Nvnc $x = \frac{p}{q}$ et $y = \frac{r}{s}$ , et rvrsvs $a^x = u^p$ cvm $u = \sqrt[q]{a}$ . Per definitionem, $\left(a^x\right)^y$ est radix $s$ -ta nvmeri $\left(u^p\right)^r$ ; hvnc nvmervm positivvm appellamvs $w$ .
$w^{qs} = \left(\resaltar{w^s}\right)^q = \left(u^{pr}\right)^q = \left(\resaltar{u^q}\right)^{pr} = \dato{wp}{a^{pr}}$
Elevamvs $w$ ad potentiam $qs$ , privs exponentem $s$ conivngendo, qvi radicem tollit, et deinde exponentem $q$ , cvm $u^q = a$ .
$\left(a^x\right)^y = w = \sqrt[qs]{\resaltar{a^{pr}}} = \resaltar{a^{\frac{pr}{qs}}} = a^{xy}$
Per vnicitatem radicis principalis, $w$ est radix indicis $qs$ nvmeri $a^{pr}$ , qvae est potentia exponentis $\frac{pr}{qs} = xy$ .

Ex prima lege conseqventiae seqvvntvr: cvm $a^x \cdot a^{-x} = a^0 = 1$ , nvmervs $a^{-x}$ est inversvm nvmeri $a^x$ , et tvm $\frac{a^x}{a^y} = a^x \cdot a^{-y} = a^{x - y}$ . Leges sinvnt cvm exponentibvs rationalibvs calcvlare vt cvm integris. Exempli gratia:

\sqrt{2} \cdot \sqrt[3]{2} = 2^{\frac{1}{2}} \cdot 2^{\frac{1}{3}} = 2^{\frac{5}{6}} = \sqrt[6]{32}, \qquad \left(\sqrt{2}\right)^{\frac{2}{3}} = 2^{\frac{1}{2} \cdot \frac{2}{3}} = \sqrt[3]{2}

Potentiae exponentis rationalis et ordo

In lectione de generibvs nvmerorvm demonstratvm est, et in lectione de recta reali ad inaeqvalitates adhibitvm, potentias exponentis natvralis ordinem nvmerorvm non negativorvm servare: si $0 \leq u < v$ , tvm $u^n < v^n$ . Potentiae exponentis rationalis hanc proprietatem respectv basis hereditate accipivnt, et aliam respectv exponentis addvnt, cvivs sensvs pendet ex eo qvod basis maior an minor sit qvam $1$ .

LemmaMonotonia potentiarvm rationalivm

Sint $x < y$ rationales. Si $a > 1$ , tvm $a^x < a^y$ ; si $0 < a < 1$ , tvm $a^x > a^y$ . Praeterea, si $x > 0$ et $0 < s < t$ , tvm $s^x < t^x$ . Figvra delineationes ostendit fvnctionvm $y = t^{\frac{1}{3}}$ , $y = t^{\frac{1}{2}}$ et $y = t^{\frac{3}{2}}$ pro $t > 0$ : ad dextram nvmeri $t = 1$ exponens maior valorem maiorem dat, ad sinistram contrarivm accidit, et in $t = 1$ tres cvrvae se secant.

Demonstratio

$x = \frac{p}{q}, \quad y = \frac{r}{q}, \quad p < r, \quad \dato{u1}{u > 1}$
Sit $a > 1$ . Scribimvs $x$ et $y$ cvm denominatore commvni $q$ , et sit $u = \sqrt[q]{a}$ . Tvm $u > 1$ : si esset $u \leq 1$ , per monotoniam potentiarvm haberetvr $u^q \leq 1$ , contra $u^q = a > 1$ .
$a^y = u^r = \resaltar{u^p \cdot u^{r - p}} > \resaltar{u^p} = a^x$
Cvm $r - p$ natvralis sit, $u^{r - p} > 1^{r - p} = 1$ , et mvltiplicatio per positivvm $u^p$ inaeqvalitatem servat.
$\frac{1}{a^x} < \frac{1}{a^y} \rightarrow \resaltar{a^x > a^y}$
Si $0 < a < 1$ , inversvm eivs maivs est qvam $1$ , et per svperiora $\left(\frac{1}{a}\right)^x < \left(\frac{1}{a}\right)^y$ . Cvm $\left(\frac{1}{a}\right)^x = \frac{1}{a^x}$ per leges exponentivm, svmptis inversis dvorvm positivorvm inaeqvalitas invertitvr.
$1^x = 1$
Si $a = 1$ , omnes eivs potentiae valent $1$ , qvia radices et potentiae nvmeri $1$ svnt $1$ . Ideo tres delineationes per pvnctvm $(1, 1)$ transevnt.
$0 < s < t \rightarrow \resaltar{s^p < t^p} \rightarrow \resaltar{\sqrt[q]{s^p} < \sqrt[q]{t^p}}$
Pro monotonia in basi, sit $x = \frac{p}{q}$ cvm $p$ natvrali. Ex $s < t$ seqvitvr $s^p < t^p$ ; et si esset $\sqrt[q]{s^p} \geq \sqrt[q]{t^p}$ , elevando ad $q$ fieret $s^p \geq t^p$ , qvod falsvm est.

Delineationes fvnctionvm potentiae accvrate in lectione de fvnctionibvs algebraicis tractabvntvr; hic dvae monotoniae svfficivnt, qvae ad nvmeros comparandos adhibebvntvr.

In reliqva parte hvivs lectionis

Leges radicalivm
Radix svmmae
Simplificatio radicalivm
Denominatores rationales reddendi
Comparatio nvmerorvm sine machinvla calcvlatoria
Radicalia dvplicia
Problemata solvta
Series radicvm
Denominator trivm terminorvm
Denominator cvm radicibvs cvbicis
Svmma radicalivm dvplicivm
Svmma radicvm cvbicarvm qvae rationalis est
Problemata proposita
Dvae radices qvarvm svmma constans est
Dvae differentiae radicvm
Forma foliorvm chartae
Potentiae differentiae radicvm
Radix cvbica qvae radicibvs qvadratis non scribitvr