Finite Mathematik Beispiele

\sqrt{m^{7} n^{11}} - \sqrt{m^{11} n^{5}} + \sqrt{m^{5} n}

$\sqrt{m^{7} n^{11}} - \sqrt{m^{11} n^{5}} + \sqrt{m^{5} n}$

Schritt 1

Benutze

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ , um

\sqrt{m^{7} n^{11}}

$\sqrt{m^{7} n^{11}}$ als

{(m^{7} n^{11})}^{\frac{1}{2}}

${(m^{7} n^{11})}^{\frac{1}{2}}$ neu zu schreiben.

{(m^{7} n^{11})}^{\frac{1}{2}} - \sqrt{m^{11} n^{5}} + \sqrt{m^{5} n}

${(m^{7} n^{11})}^{\frac{1}{2}} - \sqrt{m^{11} n^{5}} + \sqrt{m^{5} n}$

Schritt 2

Benutze

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ , um

\sqrt{m^{11} n^{5}}

$\sqrt{m^{11} n^{5}}$ als

{(m^{11} n^{5})}^{\frac{1}{2}}

${(m^{11} n^{5})}^{\frac{1}{2}}$ neu zu schreiben.

{(m^{7} n^{11})}^{\frac{1}{2}} - {(m^{11} n^{5})}^{\frac{1}{2}} + \sqrt{m^{5} n}

${(m^{7} n^{11})}^{\frac{1}{2}} - {(m^{11} n^{5})}^{\frac{1}{2}} + \sqrt{m^{5} n}$

Schritt 3

Benutze

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ , um

\sqrt{m^{5} n}

$\sqrt{m^{5} n}$ als

{(m^{5} n)}^{\frac{1}{2}}

${(m^{5} n)}^{\frac{1}{2}}$ neu zu schreiben.

{(m^{7} n^{11})}^{\frac{1}{2}} - {(m^{11} n^{5})}^{\frac{1}{2}} + {(m^{5} n)}^{\frac{1}{2}}

${(m^{7} n^{11})}^{\frac{1}{2}} - {(m^{11} n^{5})}^{\frac{1}{2}} + {(m^{5} n)}^{\frac{1}{2}}$

Schritt 4

Wende die Produktregel auf

m^{7} n^{11}

$m^{7} n^{11}$ an.

{(m^{7})}^{\frac{1}{2}} {(n^{11})}^{\frac{1}{2}} - {(m^{11} n^{5})}^{\frac{1}{2}} + {(m^{5} n)}^{\frac{1}{2}}

${(m^{7})}^{\frac{1}{2}} {(n^{11})}^{\frac{1}{2}} - {(m^{11} n^{5})}^{\frac{1}{2}} + {(m^{5} n)}^{\frac{1}{2}}$

Schritt 5

Multipliziere die Exponenten in

{(m^{7})}^{\frac{1}{2}}

${(m^{7})}^{\frac{1}{2}}$ .

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Schritt 5.1

Wende die Potenzregel an und multipliziere die Exponenten,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

m^{7 (\frac{1}{2})} {(n^{11})}^{\frac{1}{2}} - {(m^{11} n^{5})}^{\frac{1}{2}} + {(m^{5} n)}^{\frac{1}{2}}

$m^{7 (\frac{1}{2})} {(n^{11})}^{\frac{1}{2}} - {(m^{11} n^{5})}^{\frac{1}{2}} + {(m^{5} n)}^{\frac{1}{2}}$

Schritt 5.2

Kombiniere

7

$7$ und

\frac{1}{2}

$\frac{1}{2}$ .

m^{\frac{7}{2}} {(n^{11})}^{\frac{1}{2}} - {(m^{11} n^{5})}^{\frac{1}{2}} + {(m^{5} n)}^{\frac{1}{2}}

$m^{\frac{7}{2}} {(n^{11})}^{\frac{1}{2}} - {(m^{11} n^{5})}^{\frac{1}{2}} + {(m^{5} n)}^{\frac{1}{2}}$

m^{\frac{7}{2}} {(n^{11})}^{\frac{1}{2}} - {(m^{11} n^{5})}^{\frac{1}{2}} + {(m^{5} n)}^{\frac{1}{2}}

$m^{\frac{7}{2}} {(n^{11})}^{\frac{1}{2}} - {(m^{11} n^{5})}^{\frac{1}{2}} + {(m^{5} n)}^{\frac{1}{2}}$

Schritt 6

Multipliziere die Exponenten in

{(n^{11})}^{\frac{1}{2}}

${(n^{11})}^{\frac{1}{2}}$ .

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Schritt 6.1

Wende die Potenzregel an und multipliziere die Exponenten,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

m^{\frac{7}{2}} n^{11 (\frac{1}{2})} - {(m^{11} n^{5})}^{\frac{1}{2}} + {(m^{5} n)}^{\frac{1}{2}}

$m^{\frac{7}{2}} n^{11 (\frac{1}{2})} - {(m^{11} n^{5})}^{\frac{1}{2}} + {(m^{5} n)}^{\frac{1}{2}}$

Schritt 6.2

Kombiniere

11

$11$ und

\frac{1}{2}

$\frac{1}{2}$ .

m^{\frac{7}{2}} n^{\frac{11}{2}} - {(m^{11} n^{5})}^{\frac{1}{2}} + {(m^{5} n)}^{\frac{1}{2}}

$m^{\frac{7}{2}} n^{\frac{11}{2}} - {(m^{11} n^{5})}^{\frac{1}{2}} + {(m^{5} n)}^{\frac{1}{2}}$

m^{\frac{7}{2}} n^{\frac{11}{2}} - {(m^{11} n^{5})}^{\frac{1}{2}} + {(m^{5} n)}^{\frac{1}{2}}

$m^{\frac{7}{2}} n^{\frac{11}{2}} - {(m^{11} n^{5})}^{\frac{1}{2}} + {(m^{5} n)}^{\frac{1}{2}}$

Schritt 7

Wende die Produktregel auf

m^{11} n^{5}

$m^{11} n^{5}$ an.

m^{\frac{7}{2}} n^{\frac{11}{2}} - ({(m^{11})}^{\frac{1}{2}} {(n^{5})}^{\frac{1}{2}}) + {(m^{5} n)}^{\frac{1}{2}}

$m^{\frac{7}{2}} n^{\frac{11}{2}} - ({(m^{11})}^{\frac{1}{2}} {(n^{5})}^{\frac{1}{2}}) + {(m^{5} n)}^{\frac{1}{2}}$

Schritt 8

Multipliziere die Exponenten in

{(m^{11})}^{\frac{1}{2}}

${(m^{11})}^{\frac{1}{2}}$ .

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Schritt 8.1

Wende die Potenzregel an und multipliziere die Exponenten,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

m^{\frac{7}{2}} n^{\frac{11}{2}} - (m^{11 (\frac{1}{2})} {(n^{5})}^{\frac{1}{2}}) + {(m^{5} n)}^{\frac{1}{2}}

$m^{\frac{7}{2}} n^{\frac{11}{2}} - (m^{11 (\frac{1}{2})} {(n^{5})}^{\frac{1}{2}}) + {(m^{5} n)}^{\frac{1}{2}}$

Schritt 8.2

Kombiniere

11

$11$ und

\frac{1}{2}

$\frac{1}{2}$ .

m^{\frac{7}{2}} n^{\frac{11}{2}} - (m^{\frac{11}{2}} {(n^{5})}^{\frac{1}{2}}) + {(m^{5} n)}^{\frac{1}{2}}

$m^{\frac{7}{2}} n^{\frac{11}{2}} - (m^{\frac{11}{2}} {(n^{5})}^{\frac{1}{2}}) + {(m^{5} n)}^{\frac{1}{2}}$

m^{\frac{7}{2}} n^{\frac{11}{2}} - (m^{\frac{11}{2}} {(n^{5})}^{\frac{1}{2}}) + {(m^{5} n)}^{\frac{1}{2}}

$m^{\frac{7}{2}} n^{\frac{11}{2}} - (m^{\frac{11}{2}} {(n^{5})}^{\frac{1}{2}}) + {(m^{5} n)}^{\frac{1}{2}}$

Schritt 9

Multipliziere die Exponenten in

{(n^{5})}^{\frac{1}{2}}

${(n^{5})}^{\frac{1}{2}}$ .

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Schritt 9.1

Wende die Potenzregel an und multipliziere die Exponenten,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

m^{\frac{7}{2}} n^{\frac{11}{2}} - (m^{\frac{11}{2}} n^{5 (\frac{1}{2})}) + {(m^{5} n)}^{\frac{1}{2}}

$m^{\frac{7}{2}} n^{\frac{11}{2}} - (m^{\frac{11}{2}} n^{5 (\frac{1}{2})}) + {(m^{5} n)}^{\frac{1}{2}}$

Schritt 9.2

Kombiniere

5

$5$ und

\frac{1}{2}

$\frac{1}{2}$ .

m^{\frac{7}{2}} n^{\frac{11}{2}} - (m^{\frac{11}{2}} n^{\frac{5}{2}}) + {(m^{5} n)}^{\frac{1}{2}}

$m^{\frac{7}{2}} n^{\frac{11}{2}} - (m^{\frac{11}{2}} n^{\frac{5}{2}}) + {(m^{5} n)}^{\frac{1}{2}}$

m^{\frac{7}{2}} n^{\frac{11}{2}} - (m^{\frac{11}{2}} n^{\frac{5}{2}}) + {(m^{5} n)}^{\frac{1}{2}}

$m^{\frac{7}{2}} n^{\frac{11}{2}} - (m^{\frac{11}{2}} n^{\frac{5}{2}}) + {(m^{5} n)}^{\frac{1}{2}}$

Schritt 10

Entferne die Klammern.

m^{\frac{7}{2}} n^{\frac{11}{2}} - m^{\frac{11}{2}} n^{\frac{5}{2}} + {(m^{5} n)}^{\frac{1}{2}}

$m^{\frac{7}{2}} n^{\frac{11}{2}} - m^{\frac{11}{2}} n^{\frac{5}{2}} + {(m^{5} n)}^{\frac{1}{2}}$

Schritt 11

Wende die Produktregel auf

m^{5} n

$m^{5} n$ an.

m^{\frac{7}{2}} n^{\frac{11}{2}} - m^{\frac{11}{2}} n^{\frac{5}{2}} + {(m^{5})}^{\frac{1}{2}} n^{\frac{1}{2}}

$m^{\frac{7}{2}} n^{\frac{11}{2}} - m^{\frac{11}{2}} n^{\frac{5}{2}} + {(m^{5})}^{\frac{1}{2}} n^{\frac{1}{2}}$

Schritt 12

Multipliziere die Exponenten in

{(m^{5})}^{\frac{1}{2}}

${(m^{5})}^{\frac{1}{2}}$ .

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Schritt 12.1

Wende die Potenzregel an und multipliziere die Exponenten,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

m^{\frac{7}{2}} n^{\frac{11}{2}} - m^{\frac{11}{2}} n^{\frac{5}{2}} + m^{5 (\frac{1}{2})} n^{\frac{1}{2}}

$m^{\frac{7}{2}} n^{\frac{11}{2}} - m^{\frac{11}{2}} n^{\frac{5}{2}} + m^{5 (\frac{1}{2})} n^{\frac{1}{2}}$

Schritt 12.2

Kombiniere

5

$5$ und

\frac{1}{2}

$\frac{1}{2}$ .

m^{\frac{7}{2}} n^{\frac{11}{2}} - m^{\frac{11}{2}} n^{\frac{5}{2}} + m^{\frac{5}{2}} n^{\frac{1}{2}}

$m^{\frac{7}{2}} n^{\frac{11}{2}} - m^{\frac{11}{2}} n^{\frac{5}{2}} + m^{\frac{5}{2}} n^{\frac{1}{2}}$

m^{\frac{7}{2}} n^{\frac{11}{2}} - m^{\frac{11}{2}} n^{\frac{5}{2}} + m^{\frac{5}{2}} n^{\frac{1}{2}}

$m^{\frac{7}{2}} n^{\frac{11}{2}} - m^{\frac{11}{2}} n^{\frac{5}{2}} + m^{\frac{5}{2}} n^{\frac{1}{2}}$

Schritt 13

Schreibe

m^{\frac{7}{2}} n^{\frac{11}{2}} - m^{\frac{11}{2}} n^{\frac{5}{2}} + m^{\frac{5}{2}} n^{\frac{1}{2}}

$m^{\frac{7}{2}} n^{\frac{11}{2}} - m^{\frac{11}{2}} n^{\frac{5}{2}} + m^{\frac{5}{2}} n^{\frac{1}{2}}$ in eine faktorisierte Form um.

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Schritt 13.1

Faktorisiere

m^{\frac{5}{2}} n^{\frac{1}{2}}

$m^{\frac{5}{2}} n^{\frac{1}{2}}$ aus

m^{\frac{7}{2}} n^{\frac{11}{2}} - m^{\frac{11}{2}} n^{\frac{5}{2}} + m^{\frac{5}{2}} n^{\frac{1}{2}}

$m^{\frac{7}{2}} n^{\frac{11}{2}} - m^{\frac{11}{2}} n^{\frac{5}{2}} + m^{\frac{5}{2}} n^{\frac{1}{2}}$ heraus.

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Schritt 13.1.1

Faktorisiere

m^{\frac{5}{2}} n^{\frac{1}{2}}

$m^{\frac{5}{2}} n^{\frac{1}{2}}$ aus

m^{\frac{7}{2}} n^{\frac{11}{2}}

$m^{\frac{7}{2}} n^{\frac{11}{2}}$ heraus.

m^{\frac{5}{2}} n^{\frac{1}{2}} (m^{\frac{2}{2}} n^{\frac{10}{2}}) - m^{\frac{11}{2}} n^{\frac{5}{2}} + m^{\frac{5}{2}} n^{\frac{1}{2}}

$m^{\frac{5}{2}} n^{\frac{1}{2}} (m^{\frac{2}{2}} n^{\frac{10}{2}}) - m^{\frac{11}{2}} n^{\frac{5}{2}} + m^{\frac{5}{2}} n^{\frac{1}{2}}$

Schritt 13.1.2

Faktorisiere

m^{\frac{5}{2}} n^{\frac{1}{2}}

$m^{\frac{5}{2}} n^{\frac{1}{2}}$ aus

- m^{\frac{11}{2}} n^{\frac{5}{2}}

$- m^{\frac{11}{2}} n^{\frac{5}{2}}$ heraus.

m^{\frac{5}{2}} n^{\frac{1}{2}} (m^{\frac{2}{2}} n^{\frac{10}{2}}) + m^{\frac{5}{2}} n^{\frac{1}{2}} (- m^{\frac{6}{2}} n^{\frac{4}{2}}) + m^{\frac{5}{2}} n^{\frac{1}{2}}

$m^{\frac{5}{2}} n^{\frac{1}{2}} (m^{\frac{2}{2}} n^{\frac{10}{2}}) + m^{\frac{5}{2}} n^{\frac{1}{2}} (- m^{\frac{6}{2}} n^{\frac{4}{2}}) + m^{\frac{5}{2}} n^{\frac{1}{2}}$

Schritt 13.1.3

Faktorisiere

m^{\frac{5}{2}} n^{\frac{1}{2}}

$m^{\frac{5}{2}} n^{\frac{1}{2}}$ aus

m^{\frac{5}{2}} n^{\frac{1}{2}}

$m^{\frac{5}{2}} n^{\frac{1}{2}}$ heraus.

m^{\frac{5}{2}} n^{\frac{1}{2}} (m^{\frac{2}{2}} n^{\frac{10}{2}}) + m^{\frac{5}{2}} n^{\frac{1}{2}} (- m^{\frac{6}{2}} n^{\frac{4}{2}}) + m^{\frac{5}{2}} n^{\frac{1}{2}} (1)

$m^{\frac{5}{2}} n^{\frac{1}{2}} (m^{\frac{2}{2}} n^{\frac{10}{2}}) + m^{\frac{5}{2}} n^{\frac{1}{2}} (- m^{\frac{6}{2}} n^{\frac{4}{2}}) + m^{\frac{5}{2}} n^{\frac{1}{2}} (1)$

Schritt 13.1.4

Faktorisiere

m^{\frac{5}{2}} n^{\frac{1}{2}}

$m^{\frac{5}{2}} n^{\frac{1}{2}}$ aus

m^{\frac{5}{2}} n^{\frac{1}{2}} (m^{\frac{2}{2}} n^{\frac{10}{2}}) + m^{\frac{5}{2}} n^{\frac{1}{2}} (- m^{\frac{6}{2}} n^{\frac{4}{2}})

$m^{\frac{5}{2}} n^{\frac{1}{2}} (m^{\frac{2}{2}} n^{\frac{10}{2}}) + m^{\frac{5}{2}} n^{\frac{1}{2}} (- m^{\frac{6}{2}} n^{\frac{4}{2}})$ heraus.

m^{\frac{5}{2}} n^{\frac{1}{2}} (m^{\frac{2}{2}} n^{\frac{10}{2}} - m^{\frac{6}{2}} n^{\frac{4}{2}}) + m^{\frac{5}{2}} n^{\frac{1}{2}} (1)

$m^{\frac{5}{2}} n^{\frac{1}{2}} (m^{\frac{2}{2}} n^{\frac{10}{2}} - m^{\frac{6}{2}} n^{\frac{4}{2}}) + m^{\frac{5}{2}} n^{\frac{1}{2}} (1)$

Schritt 13.1.5

Faktorisiere

m^{\frac{5}{2}} n^{\frac{1}{2}}

$m^{\frac{5}{2}} n^{\frac{1}{2}}$ aus

$m^{\frac{5}{2}} n^{\frac{1}{2}} (m^{\frac{2}{2}} n^{\frac{10}{2}} - m^{\frac{6}{2}} n^{\frac{4}{2}}) + m^{\frac{5}{2}} n^{\frac{1}{2}} (1)$ heraus.

$m^{\frac{5}{2}} n^{\frac{1}{2}} (m^{\frac{2}{2}} n^{\frac{10}{2}} - m^{\frac{6}{2}} n^{\frac{4}{2}} + 1)$

Schritt 13.2

Dividiere

$2$ durch

$2$ .

$m^{\frac{5}{2}} n^{\frac{1}{2}} (m^{1} n^{\frac{10}{2}} - m^{\frac{6}{2}} n^{\frac{4}{2}} + 1)$

Schritt 13.3

Vereinfache.

$m^{\frac{5}{2}} n^{\frac{1}{2}} (m n^{\frac{10}{2}} - m^{\frac{6}{2}} n^{\frac{4}{2}} + 1)$

Schritt 13.4

Dividiere

$10$ durch

$2$ .

$m^{\frac{5}{2}} n^{\frac{1}{2}} (m n^{5} - m^{\frac{6}{2}} n^{\frac{4}{2}} + 1)$

Schritt 13.5

Dividiere

$6$ durch

$2$ .

$m^{\frac{5}{2}} n^{\frac{1}{2}} (m n^{5} - m^{3} n^{\frac{4}{2}} + 1)$

Schritt 13.6

Dividiere

$4$ durch

$2$ .

$m^{\frac{5}{2}} n^{\frac{1}{2}} (m n^{5} - m^{3} n^{2} + 1)$