Compare Powers: (3/10)^7 vs 0.3^8 - Which Value is Larger?

Which is larger?

$(\frac{3}{10})^7\text{ }_{——}(0.3)^8$

To determine which of the two expressions is larger, we need to evaluate them using a comparability method:

• Step 1: Observe that these are powers of a common base. We first convert $(0.3)^8$ to a fraction form. The number $0.3$ is equivalent to $\frac{3}{10}$.
• Step 2: Thus, $(0.3)^8$ becomes $\left(\frac{3}{10}\right)^8$.
• Step 3: We then compare $\left(\frac{3}{10}\right)^7$ and $\left(\frac{3}{10}\right)^8$. Since both have the same base, $\frac{3}{10}$, we compare the exponents directly: $7$ and $8$.
• Step 4: Clearly, any number to the power of 7 will be larger than the same number to the power of 8, since raising a positive fraction to a higher power results in a smaller number.

This analysis shows that $\left(\frac{3}{10}\right)^7$ is larger than $(0.3)^8$. Therefore, using the symbol for comparison:

The correct comparison is $(\frac{3}{10})^7 > (0.3)^8$.

Accordingly, the solution to this problem as stated is incorrect in the given answer. The correct comparison should indeed be:

$(\frac{3}{10})^7 > (0.3)^8$