Basic on law of exponents, quotient rule,power of product rule, powerful of quotation rule

Here are the explanations for the laws of exponents you mentioned:

1. Quotient Rule
This rule states that when dividing two powers with the same base, you subtract the exponents.

$$\frac{a^m}{a^n} = a^{m-n}$$

Example:

$$\frac{x^7}{x^3} = x^{7-3} = x^4$$

The quotient rule is:

$$\boxed{\frac{a^m}{a^n} = a^{m-n}}$$

2. Power of a Product Rule
This rule states that when a product of bases is raised to an exponent, you apply the exponent to each base in the product.

$$(ab)^n = a^n b^n$$

Example:

$$(2x)^3 = 2^3 x^3 = 8x^3$$

The power of a product rule is:

$$\boxed{(ab)^n = a^n b^n}$$

3. Power of a Quotient Rule
This rule states that when a quotient (fraction) is raised to an exponent, you apply the exponent to both the numerator and the denominator.

$$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$

Example:

$$\left(\frac{y}{z}\right)^4 = \frac{y^4}{z^4}$$

The power of a quotient rule is:

$$\boxed{\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}}$$