Solve (4²/7⁴)²: Calculating Powers of Complex Fractions

When you raise a power to another power, you're essentially multiplying the base by itself multiple times. For example, $(4^2)^2$ means $4^2 \times 4^2$, which by the multiplication rule becomes $4^{2+2} = 4^4$. The power rule $(a^m)^n = a^{m \times n}$ is a shortcut for this process!

Yes! When you have $(\frac{a}{b})^n$, the exponent applies to the entire fraction. This means $(\frac{4^2}{7^4})^2 = \frac{(4^2)^2}{(7^4)^2}$. Apply the power rule to both the top and bottom separately.

$4^2 \times 2 = 16 \times 2 = 32$, but $(4^2)^2 = 4^4 = 256$. The parentheses make all the difference! Without parentheses, you multiply the result by 2. With parentheses, you raise the entire power to the 2nd power.

The final answer $\frac{4^4}{7^8}$ cannot be simplified further because 4 and 7 share no common factors. However, you could write it as $\frac{256}{5764801}$ if you calculate the actual values, but the exponential form is preferred.

Work backwards! Start with $\frac{4^2}{7^4} = \frac{16}{2401}$, then square this fraction: $(\frac{16}{2401})^2 = \frac{16^2}{2401^2} = \frac{256}{5764801}$. This should equal $\frac{4^4}{7^8} = \frac{256}{5764801}$ ✓