Getting Started 
Misconception/Error The student does not understand radical notation. 
Examples of Student Work at this Level The student:
 RewritesÂ Â asÂ Â but is unable to correctly rewrite any of the other expressions.
 Attempts to uses a calculator to approximate each expression.
 Confuses the index with the exponent and writes Â as , Â as , or .
 Interprets the index as a coefficient and writes Â or .
 Interprets the index as a divisor and writesÂ Â orÂ .

Questions Eliciting Thinking What is ? How would you rewriteÂ Â using exponents?
What is meant by the index of a radical? What does the index mean? 
Instructional Implications If needed, review related terminology such as radical, radicand, index, exponent, base, and power. Also, review the properties of exponents. Review the meaning of the index of a radical and how to represent a radical such as Â or Â in equivalent exponential form. Be sure the student understands the convention that if no index is given, then it is two (or a square root). Ask the student to revise his or her responses to the first two questions.
Explain the definition of rational exponents and provide examples of expressions written in both radical and exponential form. Ask the student to revise his or her response to the third question. Model rewriting Â as . Provide additional examples of numerical and variable expressions written in radical form and ask the student to rewrite each in an equivalent exponential form. 
Moving Forward 
Misconception/Error The student is unable to rewrite a radical expression when theÂ radicand is written as a power. 
Examples of Student Work at this Level The student rewritesÂ Â as Â andÂ Â as . However, the student:
 RewritesÂ Â asÂ ,Â , or 18.52.
 Interchanges the index and the powerÂ and rewrites Â asÂ Â and Â as .

Questions Eliciting Thinking What does the numerator of your rational exponent mean? What does the denominator of your rational exponent mean?
How would you simplify ? What would you do first? 
Instructional Implications Review the definition of rational exponents and provide examples of expressions written in both radical and exponential form. Remind the student that , so .
Allow the student to revise his or her responses to the third and fourth problems. Provide additional examples of numerical and variable expressions written in radical form and ask the student to rewrite each in an equivalent exponential form. 
Almost There 
Misconception/Error The student is unable to rewrite radical expressions as a single power of the smallest whole number base. 
Examples of Student Work at this Level The student rewrites Â asÂ Â andÂ asÂ . Then, the studentÂ rewritesÂ Â andÂ Â in equivalent forms butÂ not as single powers of seven. The student:
 Uses a base other than seven.
 Does not write the exponent as a single power.

Questions Eliciting Thinking How are Â andÂ Â similar? How are they different?
Can you express 49 as a power of 7?Â How would you rewrite ?
How can you rewriteÂ Â as a single power of seven. What law of exponents can be applied to this expression? 
Instructional Implications Review with the student what it means to write an expression as a single power of seven. Assist the student in identifying answers not written in this form and ask the student to revise his or her work. If necessary, review the laws of exponents. Offer the studentÂ additional examples of numerical and variable expressions written in radical form and ask the student to rewrite each in an equivalent exponential form. 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level The student correctly rewrites each expression as a single power of seven.

Questions Eliciting Thinking Is it possible to avoid using radicals and always use fractional exponents? 
Instructional Implications Challenge the student to:
 Rewrite Â as a single power of two.
 Represent the square root of the cube root of x as (a) a single radical and (b) a power of x.
