Question

If n is a positive integer such that $(^7\sqrt{10})(^7\sqrt{10})^2).....(^7\sqrt{10})^n) > 999$, then the smallest value of n is

Answer 1

Given that:
$(^7\sqrt{10})(^7\sqrt{10})^2).....(^7\sqrt{10})^n) > 999$

This implies:
$10^{\frac{1}{7}} \times 10^{\frac{2}{7}} \times .... \times 10^{\frac{n}{7}} > 999$

By multiplying powers with the same base, you add the exponents:
$10^{{(\frac{1}{7} +\frac{ 2}{7} + ... + \frac{n}{7})}} > 999$
$10^{(\frac{1+2+...+n)}{7})} > 999$

Now, we know $10^3 = 1000$ and that's the closest power of 10 to 999.
So,
$10^{(\frac{1+2+...+n)}{7})} > 10^3$

For minimum value of n,

$\frac{1+2+....+n}{7}=3$

$1+2+...+n=21$
Now if n=6
$1+2+3+4+5+6=21$
That means, the smallest value for n is 6

So, the answer is 6.