simplificar (10!)/(10^k(10-k)!)

Solução

simplificar

\frac{10 !}{1 0 ^{k} ( 10 - k ) !}

\frac{567 \cdot 5 ^{2 - k} \cdot 2 ^{- k + 8}}{( 10 - k ) !}

Passos da solução

\frac{10 !}{1 0 ^{k} ( 10 - k ) !}

10! = 3628800

= \frac{3628800}{1 0 ^{k} ( 10 - k ) !}

Fatorar

3628800 : 1 0^{2} \cdot 36288

= \frac{1 0 ^{2} \cdot 36288}{1 0 ^{k} ( 10 - k ) !}

Aplicar as propriedades dos expoentes:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

\frac{1 0 ^{2}}{1 0 ^{k}} = 1 0^{2 - k}

= \frac{1 0 ^{2 - k} \cdot 36288}{( 10 - k ) !}

Simplificar

1 0^{2 - k} \cdot 36288 : 567 \cdot 5^{2 - k} \cdot 2^{- k + 8}

= \frac{567 \cdot 5 ^{2 - k} \cdot 2 ^{- k + 8}}{( 10 - k ) !}

8 1^{\frac{5}{4}}

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