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Finding Integral Part of Numbers of the Form a^b Where a Is Irrational

The value of ...

Question

The value of ${\frac{2^{4 n}}{15}}, n \in N$ is
(where {.} represents fractional part function)

A

\frac{1}{15}

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B

\frac{11}{15}

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C

\frac{4}{15}

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D

\frac{14}{15}

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Solution

The correct option is A $\frac{1}{15}$
$2^{4 n} = 16^{n} = (1 + 15)^{n} = 1 +^{n} C_{1} \cdot 15 +^{n} C_{2} \cdot 15^{2} + \dots +^{n} C_{n} \cdot 15^{n} = 1 + 15 k, k \in N$

Now,
$\frac{2^{4 n}}{15} = k + \frac{1}{15}$

Hence, ${\frac{2^{4 n}}{15}} = \frac{1}{15}$

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Finding Integral Part of Numbers of the Form a^b Where a Is Irrational

Standard XII Mathematics

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