On the number line, ( \sqrt{\frac{81}{196}} ) is equal to which point?

Correct answer: ( \frac{9}{14} )

The expression is \\(\sqrt{\frac{81}{196}}\\). Since both 81 and 196 are perfect squares, take their positive square roots: \\(\sqrt{81}=9\\) and \\(\sqrt{196}=14\\). Therefore \\(\sqrt{\frac{81}{196}}=\frac{\sqrt{81}}{\sqrt{196}}=\frac{9}{14}\\). The principal square root is nonnegative, so the result is positive. Hence option A represents the correct point on the number line. Options such as \\(\frac{81}{14}\\) or \\(\frac{9}{196}\\) do not result from taking the square root of numerator and denominator.

The fraction is already in lowest terms because 9 and 14 have no common factor greater than 1. Squaring the answer also verifies it: \\(\left(\frac{9}{14}\right)^2=\frac{81}{196}\\). Thus the point corresponding to the given radical is exactly \\(\frac{9}{14}\\), not a negative value or an unreduced expression. The supplied answer A is mathematically consistent and follows from the square-root property for positive numerator and denominator.