Apply the properties of exponents including: multiplying, dividing, power to a power, zero exponent, negative exponents, and turning fractional exponents into radicals. (copy)

The Basics: Multiplication and Division (00:00 - 01:20)

• Explains the product rule, where multiplying powers with the same base requires adding the exponents together ($x^a imes x^b = x^{a+b}$).

• Discusses the quotient rule, where dividing powers with the same base involves subtracting the exponent in the denominator from the exponent in the numerator ($\frac{x^a}{x^b} = x^{a-b}$).

Power to a Power Rule (01:20 - 02:05)

• Clarifies the distinction between adding and multiplying exponents. When a power is raised to another power, the exponents are multiplied ($(x^a)^b = x^{ab}$).

The Zero and Negative Exponent Rules (02:05 - 03:15)

• Provides a logical proof for why any non-zero base raised to the power of zero is always $1$ using the quotient rule.

• Explains negative exponents as a reciprocal relationship, where a negative power signifies the base belongs on the opposite side of the fraction bar ($x^{-n} = rac{1}{x^n}$).

Fractional Exponents and Radicals (03:15 - 04:00)

• Details how to convert rational exponents into radical form. The numerator acts as the power and the denominator acts as the root index ($x^{m/n} = ext{n-th root of } x^m$).