The given expression is:

$\left( -7^{12} \right)^{-3}$

We are asked to simplify this expression using the laws of exponents.

Steps:

1. Apply the power rule for exponents: $\left( a^m \right)^n = a^{m \cdot n}$ Applying this rule to the expression: $\left( -7^{12} \right)^{-3} = -7^{12 \cdot (-3)} = -7^{-36}$

2. Simplify further using the negative exponent rule: The negative exponent rule states that: $a^{-n} = \frac{1}{a^n}$ Thus: $-7^{-36} = -\frac{1}{7^{36}}$

The simplified expression is:

$-\frac{1}{7^{36}}$

Let me know if you need further clarification or if you'd like to explore additional concepts related to this problem!

Here are 5 related questions to deepen your understanding:

1. How do you simplify an expression with multiple exponents, like $(a^m)^n$?
2. What happens when the base of the exponent is negative, such as in $(-a)^m$?
3. How do you simplify expressions like $(a^m \cdot b^n)^p$?
4. What is the significance of the zero exponent in expressions like $a^0$?
5. How do you convert between exponential notation and radical notation?

Tip: Always remember that negative exponents represent the reciprocal of the base raised to the corresponding positive exponent!