To simplify the expression $x^{-7} \cdot x^5$, you can use the rule of exponents that states:

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

Applying this rule here:

$x^{-7} \cdot x^5 = x^{-7 + 5} = x^{-2}$

Since $x^{-2}$ can also be written as a positive exponent, the final answer is:

$\frac{1}{x^2}$

Would you like further details or have any questions?

Here are some related questions for further exploration:

1. How would you simplify $x^{-3} \cdot x^4$?
2. What is the rule for dividing exponents with the same base, such as $x^7 \div x^3$?
3. How does the power rule apply when raising a power to another power, such as $(x^2)^3$?
4. Can you explain what $x^0$ equals and why?
5. How would you simplify an expression with multiple variables, such as $x^3 y^{-2} \cdot x^{-1} y^4$?

Tip: Remember that negative exponents indicate the reciprocal of the base raised to the corresponding positive exponent.