Lesson 5

Negative Rational Exponents

Lesson Narrative

In this lesson, students build on what they know about positive fractional exponents, exponent rules, and graphs in order to make sense of negative fractional exponents. Like they did with positive fractional exponents, students graph $$y=2^x$$ for negative integer values of $$x$$, sketch the continuous curve between those points, and estimate the $$y$$-coordinates on the curve for various negative rational $$x$$-coordinates. This leads to the observation that expressions like $$b^{\text-x}$$ can be rewritten as $$\dfrac{1}{b^x}$$ for any rational number $$x$$.

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).

Learning Goals

Teacher Facing

• Draw a graph representing negative rational exponents and use it to estimate values.
• Use both radicals and exponents to represent numbers.

Student Facing

• Let’s investigate negative exponents.

Student Facing

• I can interpret exponents that are negative fractions.

Building On