Compare Powers: (10×3)^7 and the Reciprocal of (15×2)^-7

To solve this problem, we'll follow these steps:

• Step 1: Simplify both expressions using exponent rules.
• Step 2: Compare the simplified results to determine the appropriate sign.

First, simplify $(10 \times 3)^7$:
According to the power of a product rule, $(ab)^n = a^n \times b^n$. So,

$(10 \times 3)^7 = 10^7 \times 3^7$.

Now, simplify $\frac{1}{(15 \times 2)^{-7}}$:
Firstly, address the negative exponent: $(ab)^{-n} = \frac{1}{a^n \times b^n}$, so we have:

$(15 \times 2)^{-7} = \frac{1}{15^7 \times 2^7}$.
Then, taking the reciprocal because of the double negative (when taking the reciprocal of inverse due to $-n$),

$\frac{1}{(15 \times 2)^{-7}} = 15^7 \times 2^7$.

Now, compare the expressions:
Since $10 = 2 \times 5$ and $15 = 3 \times 5$, consider breaking each base into prime factors:

$10^7 \times 3^7 = (2^7 \times 5^7) \times 3^7$,
$15^7 \times 2^7 = (3^7 \times 5^7) \times 2^7$.

Both $10^7 \times 3^7$ and $15^7 \times 2^7$ resolve to the same product since they are permutations of the same multiplication.

Thus, we conclude:

The two expressions are equal, so the compatible sign is $=$.

Therefore, the solution to the problem is $=$.