Algebra 2: Properties of Exponents

Study Notes

Product Property of Exponents: States that when multiplying two exponential terms with the same base, the exponents are added: ( a^m * a^n = a^{m+n} ).
Quotient Property of Exponents: Describes the operation of division for exponential terms with the same base, where the exponents are subtracted: ( \frac{a^m}{a^n} = a^{m-n} ).
Definition of Negative Exponents: Indicates how to express negative exponents; a negative exponent represents the reciprocal: ( a^{-n} = \frac{1}{a^n} ) and ( \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n} ).
Zero Exponents: States that any base raised to the power of zero equals one: ( a^0 = 1 ).
Power of a Power Property: Relates to raising an exponent to another power, which results in multiplying the exponents: ( (a^m)^n = a^{mn} ).
Power of a Product Property: Describes the distribution of an exponent over a product of bases: ( (ab)^m = a^m b^m ).
Power Property of Equality: States that if two values are equal, their powers are also equal: If ( a = b ), then ( a^n = b^n ).
Common Base Property of Equality: Indicates that if two exponential expressions with the same base are equal, then their exponents must also be equal: If ( a^n = a^m ) and ( a \neq 1 ), then ( n = m ).
Power of a Quotient Property: Outlines how to apply an exponent to a fraction, distributing the exponent to both the numerator and denominator: ( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} ).