$If the number {1.2.2}^{2} {.2}^{3} {.2}^{4} . . . . . . . . . . {.2}^{20} . {3.3}^{2} {.3}^{3} {.3}^{4} . . . . . . . {.3}^{10}$
$is written in power of '6' then the highest$
$power of '6' is _____.$

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Solution

2 and 3 are factors of 6.

In the given problem the highest power of 2 is 20, whereas the highest power of 3 is 10.

The highest power of 6 will depend on the lowest power of 2 and 3.

$(2 \times 3) \times (2 \times 3)^{2} \times (2 \times 3)^{3} \times (2 \times 3)^{4} \times (2 \times 3)^{5}$

$\times (2 \times 3)^{6} \times (2 \times 3)^{7} \times (2 \times 3)^{8} \times (2 \times 3)^{9} \times (2 \times 3)^{10}$

= $6 \times 6^{2} \times 6^{3} . . . . . . . . . . . {.6}^{10}$

Using the product law of indices, we get $(6)^{1 + 2 + 3 + 4 + . . .10}$

Therefore the highest power of 6 = 1+2+3+4+5+6+7+8+9+10 = 55.

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If the number 1.2.22.23.24...........220.3.32.33.34........310 is written in power of '6' then the highest power of '6' is _____.

$If the number {1.2.2}^{2} {.2}^{3} {.2}^{4} . . . . . . . . . . {.2}^{20} . {3.3}^{2} {.3}^{3} {.3}^{4} . . . . . . . {.3}^{10}$
$is written in power of '6' then the highest$
$power of '6' is _____.$