Express ${[(2^{1 / 2} . 4^{3 / 4} . 8^{5 / 6} . 16^{7 / 8} . 32^{9 / 10})^{4}]}^{8 / 25}$ as an integer.

8192

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32768

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65536

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131072

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Solution

The correct option is B 65536
$[(2^{1 / 2} {.4}^{3 / 4} {.8}^{5 / 6} {.16}^{7 / 8} {.32}^{9 / 10})^{4}]^{8 / 25}$

$\Rightarrow$ $[\sqrt{2} . (\sqrt{2})^{2} . (\sqrt{2})^{5} . (\sqrt{2})^{7} . (\sqrt{2})^{8})^{4}]^{8 / 25}$

$\Rightarrow$ $[((\sqrt{2})^{1 + 3 + 5 + 7 + 9})^{4}]^{8 / 25}$ $[x^{m} . x^{n} = x^{m + n}]$

$\Rightarrow$ $[((\sqrt{2})^{25})^{4}]^{8 / 25}$

$\Rightarrow$ $[(\sqrt{2})^{100}]^{8 / 25}$

$\Rightarrow$ $(\sqrt{2})^{32}$

$\Rightarrow$ $((\sqrt{2})^{16})^{2}$

$\Rightarrow$ $(256)^{2}$

$\Rightarrow$ $65536$

$∴$ $[(2^{1 / 2} {.4}^{3 / 4} {.8}^{5 / 6} {.16}^{7 / 8} {.32}^{9 / 10})^{4}]^{8 / 25} = 65536$

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Q. The units digit of the result of the expression (25×25) + (21×21) + (12×12) is

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Express [(21/2.43/4.85/6.167/8.329/10)4]8/25 as an integer.

The correct option is B 65536[(21/2.43/4.85/6.167/8.329/10)4]8/25⇒ [√2.(√2)2.(√2)5.(√2)7.(√2)8)4]8/25⇒ [((√2)1+3+5+7+9)4]8/25 [xm.xn=xm+n]⇒ [((√2)25)4]8/25⇒ [(√2)100]8/25⇒ (√2)32⇒ ((√2)16)2⇒ (256)2⇒ 65536∴ [(21/2.43/4.85/6.167/8.329/10)4]8/25=65536

Express ${[(2^{1 / 2} . 4^{3 / 4} . 8^{5 / 6} . 16^{7 / 8} . 32^{9 / 10})^{4}]}^{8 / 25}$ as an integer.