### Theory:

Product law
The product law states that the exponents can be added when multiplying two powers with the same base.

$$a^{m} \times a^{n} = a^{m + n}$$, where $$a \ne 0$$ and $$a$$, $$m$$, $$n$$ are integers.
Example:
1. $$3^4 \times 3^2$$

Here, the base $$3$$ is the same for both the numbers. So, we can add the powers.

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

$$3^4 \times 3^2 = 3^{4 + 2} = 3^{6}$$

2. $$5^{-4} \times 5^{-2}$$

Method I:

$$5^{-4} \times 5^{-2}$$ $$=$$ $\frac{1}{{5}^{4}}×\frac{1}{{5}^{2}}$

$$=$$ $\frac{1}{{5}^{4}×{5}^{2}}$

$$=$$ $\frac{1}{{5}^{4+2}}$ $$=$$ $\frac{1}{{5}^{6}}$ $$=$$ $$5^{-6}$$

Thus, $$5^{-4} \times 5^{-2}$$ $$=$$ $$5^{-6}$$.

Method II:

$$5^{-4} \times 5^{-2}$$ $$=$$ $$5^{(-4)+(-2)} = 5^{-6}$$
Quotient law
The quotient law states that we can divide two powers with the same base by subtracting the exponents.

$\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}$, where $$a \ne 0$$ and $$a$$, $$m$$, $$n$$ are integers.
Example:
1. $\frac{{5}^{4}}{{5}^{2}}={5}^{4-2}={5}^{2}$

2. $\frac{{\left(-4\right)}^{6}}{{\left(-4\right)}^{-2}}={\left(-4\right)}^{6-\left(-2\right)}$

$$=$$ $$(-4)^{6+2}$$ $$=$$ $$(-4)^{8}$$

Therefore, $\frac{{\left(-4\right)}^{6}}{{\left(-4\right)}^{-2}}={\left(-4\right)}^{8}$.
Power law
The power law states that when a number is raised to a power of another power, we need to multiply the powers or exponents.

$$(a^m)^n = a^{mn}$$, where $$a \ne 0$$ and $$a$$, $$m$$, $$n$$ are integers.
Example:
1. $$(5^2)^3 = (5)^{2 \times 3} = 5^{6}$$.

2. $$[5^{(-2)}]^3 = 5^{(-2) \times 3} = (5)^{-6}$$.

3. $$[(-5)^{2}]^{3} = (-5)^{2 \times 3} = (-5)^{6}$$.

4. $$[(-5)^{2}]^{-3} = (-5)^{2 \times (-3)} = (-5)^{-6}$$.