Question

How can the quotient of powers law be used to show that a negative exponent is equivalent to the reciprocal of the positive exponent? answer in 5 sentence summary

Answer 1

The quotient of powers law states that when dividing two expressions with the same base, you subtract the exponents: \( a^m / a^n = a^{m-n} \). When we consider a negative exponent, such as \( a^{-n} \), we can express it as \( 1/a^n \) by applying the rules of exponents. This can be derived from the quotient of powers law by recognizing that \( a^{0} \) (which is equal to 1) can be rewritten as \( a^{n-n} \). Therefore, we can restate \( a^{-n} \) in terms of positive exponents: \( a^{-n} = a^{0-n} = a^0 / a^n = 1/a^n \). This shows that negative exponents are indeed equivalent to the reciprocal of the corresponding positive exponent.