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.