Student Actions

Teacher Actions

• Develop a deep and flexible understanding of the relationship between radicals and rational exponents and how they are represented in a radical expression.

• Develop strategies for problem-solving when determining an appropriate form that will be most efficient to solve mathematical problems involving both rational exponent and radical expressions.

• Develop the ability to communicate mathematically by using written and/or verbal explanations of how/why to express a radical in rational exponent form and vice versa. 

• Build procedural fluency from conceptual understanding by allowing students to discover the connection between properties of rational exponents and how rational exponents are represented in an equivalent radical form.

• Provide students with opportunities to become fluent in converting between rational exponents and radicals as well as applying properties of exponents to simplify rational exponent expressions.

• Support productive struggle by providing time and situations for students to discuss and make connections about the relationship between radical and rational expressions through various methods (ie graphing calculator, desmos)

• Pose purposeful questions that advance student understanding of the relationship between radicals and rational exponent expressions 

• Facilitate meaningful mathematical discourse by choosing tasks for students to explore and defend which form, rational exponent or radical, is most efficient to solve problems. 

Key Understandings

Misconceptions 

• Rational exponents and radicals can be converted from one to the other to make equivalent forms. 

• Operations on rational number exponents use the same properties as operations on integer value exponents.

• The numerator of a rational number exponent is transferred to the interior of the radicand to become the exponent on the base number and the denominator value becomes the root of the radicand. 

• Students may not recognize that rational exponents are radicals in a different form and vice versa.

• When multiplying or dividing rational exponents, students will not add or subtract the exponents correctly.

• Students will place the root in the numerator when converting from a radical form to a rational exponent expression.

• Students may mistake a number raised to power and then taken to a root as different from a number taken to a root and then raised to a power. Ex.  (cube root of a)⁴ = (cube root of a⁴) 

  Knowledge Connections

Prior Knowledge

Leads to 

• Use and apply rules and properties of integer exponents (PA.N.1.1)

• Writing the square roots of monomials in simplest form (A1.N.1.1) 

• Use properties of rational exponents and radicals to solve square root and cube root equations (A2.A.1.5)