Calculate (3/7)^6: Evaluating the Sixth Power of a Fraction

The problem asks us to express $\left(\frac{3}{7}\right)^6$ in another form. To solve this, we apply the exponent rule for fractions: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$.

• First, identify the numerator and the denominator in the fraction $\frac{3}{7}$.
• We have $a = 3$ and $b = 7$.
• According to the exponent rule, raise both the numerator and the denominator separately to the power of 6:

$\left(\frac{3}{7}\right)^6 = \frac{3^6}{7^6}$

This signifies that each component of the fraction is raised to the power of 6.

To verify, we compare our result with the given choices:

• Option 1: $\frac{3}{7^6}$ does not apply the exponent to the "3".
• Option 2: $\frac{3^6}{7^6}$, matches our derived expression.
• Option 3: $\frac{3^6}{7}$ does not apply the exponent to the "7".
• Option 4: $6\times\left(\frac{3}{7}\right)^5$ changes the power on the entire fraction and multiplies by 6, which is incorrect based on our interpretation.

Therefore, the solution to the problem is $\frac{3^6}{7^6}$, which corresponds to choice 2.