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:
\((\sqrt[7]{10})(\sqrt[7]{10})^2 \cdots (\sqrt[7]{10})^n) > 999\) 

This implies: 
\(10^{\frac{1}{7}} \times 10^{\frac{2}{7}} \times \cdots \times 10^{\frac{n}{7}} > 999\)

By multiplying powers with the same base, we add the exponents: 
\(10^{\left(\frac{1}{7} + \frac{2}{7} + \cdots + \frac{n}{7}\right)} > 999\)

This simplifies to: 
\(10^{\left(\frac{1+2+\cdots+n}{7}\right)} > 999\)

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

For the minimum value of \(n\), we set:
\(\frac{1+2+\cdots+n}{7} = 3\)

This implies:
\(1+2+\cdots+n = 21\)

Now, if \(n = 6\), we calculate:
\(1+2+3+4+5+6 = 21\)

This means that the smallest value for \(n\) is 6.

Answer: The smallest value of \(n\) is 6.