In this lesson, students build on what they know about positive fractional exponents, exponent rules, and graphs to make sense of negative fractional exponents. Like they did with positive rational exponents, students graph for negative integer values of , sketch the continuous curve between those points, and estimate the -coordinates on the curve for various negative rational -coordinates. This leads to the observation that expressions like can be rewritten as  for any rational number  and positive number .

There is an optional activity that goes beyond the depth of understanding required to address the standards. Use this activity to give students additional practice with rational exponents.

Students make use of structure when they use graphs to estimate the value of exponential expressions and use exponent rules to test the accuracy of their estimates (MP7).