Mastering the Fraction: Simplify (3/7)^-9

A negative exponent means "take the reciprocal" and make the exponent positive. So $(\frac{3}{7})^{-9}$ becomes $(\frac{7}{3})^9$, which equals $\frac{7^9}{3^9}$.

No! Negative exponents create reciprocals, not negative numbers. The result $\frac{7^9}{3^9}$ is positive because both $7^9$ and $3^9$ are positive.

While mathematically correct, $\frac{7^9}{3^9}$ is the preferred form because it clearly shows the power applied to both numerator and denominator separately.

Then $(\frac{7}{3})^{-9} = \frac{3^9}{7^9}$. The same rule applies - flip the fraction and make the exponent positive!

With whole numbers: $3^{-9} = \frac{1}{3^9}$. With fractions: we flip the entire fraction. Both follow the same reciprocal rule for negative exponents.