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vereinfachen (sqrt(\sqrt{64m^2*n)})^{10}

Lösung

vereinfachen

(64 m^{2} \cdot n)^{10}

32768 m^{5} n^{5}

Schritte zur Lösung

(64 m^{2} n)^{10}

Schreibe

10

um:

2 \cdot 5

= (64 m^{2} n)^{2 \cdot 5}

Wende Exponentenregel an:

a^{b c} = (a^{b})^{c},

angenommen

a \geq 0

(64 m^{2} n)^{2 \cdot 5} = ((64 m^{2} n)^{2})^{5}

= ((64 m^{2} n)^{2})^{5}

Wende Exponentenregel an:

(n a)^{n} = a,

angenommen

n i s o dd or a \geq 0

(64 m^{2} n)^{2} = 64 m^{2} n

= (64 m^{2} n)^{5}

Wende Exponentenregel an:

a^{b + c} = a^{b} \cdot a^{c}

(64 m^{2} n)^{4 + 1} = (64 m^{2} n)^{4} 64 m^{2} n

= (64 m^{2} n)^{4} 64 m^{2} n

Vereinfache

(64 m^{2} n)^{4} : ((64 m^{2} n)^{2})^{2}

= ((64 m^{2} n)^{2})^{2} 64 m^{2} n

Wende Exponentenregel an:

(n a)^{n} = a,

angenommen

n i s o dd or a \geq 0

(64 m^{2} n)^{2} = 64 m^{2} n

= (64 m^{2} n)^{2} 64 m^{2} n

Wende Radikal Regel an:

a = a^{\frac{1}{2}}

64 m^{2} n = (64 m^{2} n)^{\frac{1}{2}}

= (64 m^{2} n)^{2} (64 m^{2} n)^{\frac{1}{2}}

Wende Exponentenregel an:

a^{b} \cdot a^{c} = a^{b + c}

(64 m^{2} n)^{2} (64 m^{2} n)^{\frac{1}{2}} = (64 m^{2} n)^{2 + \frac{1}{2}}

= (64 m^{2} n)^{2 + \frac{1}{2}}

2 + \frac{1}{2} = \frac{5}{2}

= (64 m^{2} n)^{\frac{5}{2}}

Wende Exponentenregel an:

(a \cdot b)^{n} = a^{n} b^{n}

(64 m^{2} n)^{\frac{5}{2}} = 6 4^{\frac{5}{2}} (m^{2})^{\frac{5}{2}} n^{\frac{5}{2}}

= 6 4^{\frac{5}{2}} (m^{2})^{\frac{5}{2}} n^{\frac{5}{2}}

6 4^{\frac{5}{2}} = 32768

= 32768 (m^{2})^{\frac{5}{2}} n^{\frac{5}{2}}

Vereinfache

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

= 32768 m^{5} n^{\frac{5}{2}}

n^{\frac{5}{2}} = n^{5}

= 32768 m^{5} n^{5}

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