Simplify 6^-7 × 6^3: Negative and Positive Exponent Reduction

To simplify the expression $6^{-7} \times 6^3$, we use the rule for multiplying powers with the same base, which states that we add the exponents together.

Given the expression:

$6^{-7} \times 6^3$

Step 1: Identify the base and the exponents involved. The base here is 6, with exponents $-7$ and $3$.

Step 2: Apply the multiplication of powers rule:

$a^m \times a^n = a^{m+n}$

For our problem, $a = 6$, $m = -7$, and $n = 3$. Therefore:

$6^{-7} \times 6^3 = 6^{-7 + 3}$

Step 3: Calculate the exponent:

$-7 + 3 = -4$

Therefore, the expression simplifies to:

$6^{-4}$

The solution to the equation is $6^{-4}$.