107.

Stepenovanje

TEKST ZADATKA

Uprostiti izraz:

(m2−1n2)m(n+1m)n−m(n2−1m2)n(m−1n)m−n,mn=/ 0,mn=/ ±1\frac {\bigg(m^2 - {\frac 1 {n^2}}\bigg)^m \bigg(n+{\frac 1 m}\bigg)^{n-m}} {\bigg(n^2-{\frac 1 {m^2}}\bigg)^n \bigg(m-{\frac 1 n}\bigg)^{m-n}} , mn{=}\mathllap{/\,} 0, mn{=}\mathllap{/\,} \pm1
REŠENJE ZADATKA

Primeniti definiciju za sabiranje razlomaka:

(m2n2−1n2)m(mn+1m)n−m(m2n2−1m2)n(mn−1n)m−n\frac {\bigg(\frac {m^2n^2-1} {n^2}\bigg)^m \bigg(\frac {mn+1} m\bigg)^{n-m}} {\bigg(\frac {m^2n^2-1} {m^2}\bigg)^n \bigg(\frac {mn-1} n\bigg)^{m-n}}

Primeniti definiciju za razliku kvadrata:

((mn−1)(mn+1)n2)m(mn+1m)n−m((mn−1)(mn+1)m2)n(mn−1n)m−n\frac {\bigg(\frac {(mn-1)(mn+1)} {n^2}\bigg)^m \bigg(\frac {mn+1} m\bigg)^{n-m}} {\bigg(\frac {(mn-1)(mn+1)} {m^2}\bigg)^n \bigg(\frac {mn-1} n\bigg)^{m-n}}

Primeniti osnovne osobine operacija sa stepenima: (ab)m=ambm (ab)^m=a^mb^m i (ab)m=ambm \big({\frac a b}\big)^m= \frac {a^m} {b^m}

(mn−1)m(mn+1)mn2m(mn+1)n−mmn−m(mn−1)n(mn+1)nm2n(mn−1)m−nnm−n\frac {\frac {(mn-1)^m(mn+1)^m} {n^{2m}} {\frac {(mn+1)^{n-m}} {m^{n-m}}}} {\frac {(mn-1)^n(mn+1)^n} {m^{2n}} {\frac {(mn-1)^{m-n}} {n^{m-n}}}}

Pomnožiti razlomke, grupisati nepoznate i primeniti osnovnu osobinu operacija sa stepenima: am∗an=am+n a^m * a^n= a^{m+n}

(mn−1)m(mn+1)m+n−mn2mmn−m(mn−1)n+m−n(mn+1)nm2nnm−n=(mn−1)m(mn+1)nn2mmn−m(mn−1)m(mn+1)nm2nnm−n\frac {\frac {(mn-1)^m(mn+1)^{m+n-m}} {n^{2m} {m^{n-m}}}} {\frac {(mn-1)^{n+m-n}(mn+1)^n} {m^{2n}{n^{m-n}}}} = \frac {\frac {(mn-1)^m(mn+1)^n} {n^{2m} {m^{n-m}}}} {\frac {(mn-1)^m(mn+1)^n} {m^{2n}{n^{m-n}}}}

Srediti izraz:

(mn−1)m(mn+1)nn2mmn−m(mn−1)m(mn+1)nm2nnm−n=1n2mmn−m1m2nnm−n\frac {\frac {\cancel{(mn-1)^m}\cancel{(mn+1)^n}} {n^{2m} {m^{n-m}}}} {\frac {\cancel{(mn-1)^m}\cancel{(mn+1)^n}} {m^{2n}{n^{m-n}}}} = \frac {\frac 1 {n^{2m} {m^{n-m}}}} {\frac 1 {m^{2n}{n^{m-n}}}}

Srediti izraz:

m2nnm−nn2mmn−m\frac {m^{2n}n^{m-n}} {n^{2m}m^{n-m}}

Grupisati nepoznate i primeniti osnovnu osobinu operacija sa stepenima: aman=am−n \frac {a^m} {a^n}=a^{m-n}

m2n−(n−m)n(m−n)−2m=mn+mn−(n+m)m^{2n-(n-m)} n^{(m-n)-2m} = m^{n+m}n^{-(n+m)}

Primeniti definiciju stepenovanja: a−m=1am, a^{-m}={\frac 1 {a^m}} , a=/ 0 a{=}\mathllap{/\,} 0 i osnovnu osobinu operacija sa stepenima: (ab)m=ambm: \big({\frac a b}\big)^m= \frac {a^m} {b^m} :

(mn)m+n\bigg(\frac m n\bigg)^{m+n}

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