In this section, students use their understanding of exponent rules to interpret rational exponents such as as for positive numbers .

The first two lessons serve as optional review of exponent rules and the geometric meaning of square and cube roots. If students demonstrate an understanding of these concepts, the lessons can be safely skipped.

Then students look at expressions involving unit fractions in an exponent and use exponent rules to recognize the relationship between and . The understanding is expanded to other positive rational exponents and then to any rational exponent.

In this section, students are introduced to imaginary and complex numbers. After being reminded that there was a time when negative numbers were new to them and it took a while to understand how to use them, students are told that is another new kind of number. Just as the positive number line was extended to show negative values, the real number line is extended to create a complex plane to visualize imaginary and complex numbers.

Students then learn how to add and multiply complex numbers. An optional lesson is an opportunity to go beyond the standards to see how increasing powers of complex numbers follow a pattern.

It is typical to write complex numbers in the form with real numbers and . This convention is generally followed in the materials for this section. When is a real number that includes a radical, it can sometimes be convenient to write the term as (for example ) to be clear that is not included in the radical. Because the two ways of writing the terms are equivalent, either format should be accepted from students and either form can appear in the materials.

Similarly, when either or is zero, the complex number can be written as or as the single, non-zero term (for example, or ).

In this section, students return to solving quadratic equations. This time, they are able to find any real or complex solutions. Students use reasoning, completing the square, and the quadratic formula to solve the equations.

Students use the structure of the quadratic formula to help determine the number and type of solutions. Although the discriminant is mentioned, students should not be asked to memorize the expression separately from the quadratic formula. They should be able to reason about the term in the quadratic formula to determine whether solutions are real or complex.

The first lesson of the section is optional because it revisits how to complete the square and use the quadratic formula to solve quadratic equations. If students are fluent with this strategy, the lesson can be safely skipped. The last lesson of the section is optional practice in which students create their own quadratic equations so that they have the required number and type of solutions.