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

If {x} repres...

Question

If ${x}$ represents the fractional part of $x,$ then ${\frac{5^{200}}{8}}$ is

A

\frac{1}{4}

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B

\frac{1}{8}

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C

\frac{3}{8}

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D

\frac{5}{8}

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Solution

The correct option is B $\frac{1}{8}$
$\frac{5^{200}}{8} = \frac{{(5^{2})}^{200}}{8} = \frac{(25)^{100}}{8} = \frac{(1 + 24)^{100}}{8} = \frac{1 +^{100} C_{1} \cdot 24 + \dots +^{100} C_{2} (24)^{100}}{8} = \frac{1 + 24 k}{8}, k \in N = \frac{1}{8} + 3 k ∴ {\frac{5^{200}}{8}} = \frac{1}{8}$

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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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