In this section, students develop a familiarity with equations involving square and cube roots. They begin the section by recognizing that positive numbers have two square roots and recognize the benefit of understanding to represent the positive square root of . The restriction leads to a need for caution, though, when students see that solving equations involving square roots can sometimes lead to equations with solutions that are not solutions to the original equation.

After recognizing that this caution is not necessary for cube roots, students solve equations including square and cube roots.

This section ends with an optional lesson that can be used for extra practice.

Note that students have not yet been introduced to imaginary numbers, so throughout this section, the materials say that equations such as have no solution. After students encounter imaginary and complex numbers later in the unit, a distinction will be made about real solutions.

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.

	-1			0			1
(using exponents)
(equivalent expressions)		or		1		or	5

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.

In this final section, students have the opportunity to apply their thinking from throughout the unit. As this is a short section followed by an End-of-Unit Assessment, there are no section goals or checkpoint questions.