Part 1: Classify and Justify

Reference box: A rational number can be written as a fraction of integers. An irrational number cannot be written as a fraction of integers; its decimal is nonterminating and nonrepeating. A perfect square is the square of an integer, such as 25 = 5². A perfect cube is the cube of an integer, such as 27 = 3³. The radicand is the number inside a radical. The index is the small number that indicates the root: in ∛8, the index is 3. A rational exponent is an exponent written as a fraction, such as 3/4.

Reference strip: a^1/n = √[n]{a}     a^m/n = √[n]{a^m}     denominator = root index     numerator = power

a. √39 is __________________ because ________________________________________________

b. 0.9707320412481934... is __________________ because _______________________________

c. 0.5 is __________________ because __________________________________________________

d. √36 is __________________ because __________________________________________________

e. 0.519922222... is __________________ because _______________________________________

a. √25: __________________; 25 is / is not a perfect square because _____________________

b. √10: __________________; 10 is / is not a perfect square because ____________________

c. √27: __________________; 27 is / is not a perfect square because ____________________

d. ∛27: __________________; 27 is / is not a perfect cube because _____________________

e. ∛5: __________________; 5 is / is not a perfect cube because _______________________

A rational number is __________________________________________________________________

A rational exponent is _________________________________________________________________

16^1/2 = __________ is __________________, while 2^1/2 = __________ is __________________.

Sentence frame: “The exponent is rational because ____________________. The value is __________________ because ____________________.”

Part 2: Connect Representations

a. √25 = __________________________; rational or irrational: __________________

b. √10 = __________________________; rational or irrational: __________________

c. ∛27 = _________________________; rational or irrational: __________________

d. ∛5 = __________________________; rational or irrational: __________________

e. √[4]{16³} = ____________________; rational or irrational: __________________

a. 49^1/2 is __________________

b. 10^1/2 is __________________

c. 8^2/3 is __________________

d. 5^1/3 is __________________

Explanation: __________________________________________________________________________

a. Can a rational exponent produce an irrational value? Give an example and explain.

______________________________________________________________________________________

b. How do you know whether a radical is rational?

______________________________________________________________________________________

Sentence frame: “Yes / No, because a rational exponent ________________________________.”

Rational value: ____________________________ Justification: ______________________________

Irrational value: __________________________ Justification: ______________________________

Part 3: Exit Ticket and Reflection

1 — I need substantial help    2 — I need some help    3 — I am mostly confident    4 — I can explain it to someone else

One representation I can now convert is _________________________________________________

One strategy that helped me was ________________________________________________________

Answer Key — Separate Final Page

1. a. Irrational; 39 is not a perfect square. b. Irrational; the decimal is nonterminating and nonrepeating. c. Rational; it terminates and equals 1/2. d. Rational; √36 = 6. e. Rational; the decimal eventually repeats.

2. a. Rational; 25 is a perfect square. b. Irrational; 10 is not a perfect square. c. Irrational; 27 is not a perfect square. d. Rational; 27 is a perfect cube. e. Irrational; 5 is not a perfect cube.

3. A rational number is a number that can be written as a fraction of integers. A rational exponent is an exponent written as a fraction. 16^1/2 = 4, rational; 2^1/2 = √2, irrational.

4. a. 25^1/2, rational. b. 10^1/2, irrational. c. 27^1/3, rational. d. 5^1/3, irrational. e. 16^3/4, rational.

5. a. Rational, because 49 is a perfect square. b. Irrational, because 10 is not a perfect square. c. Rational, because 8 is a perfect cube and 8^2/3 = 4. d. Irrational, because 5 is not a perfect cube.

6. Yes. For example, 2^1/2 = √2, which is irrational. A radical is rational when the relevant radicand is a perfect power, such as a perfect square or perfect cube.

7. Answers vary. Example: 16^1/2 = 4 is rational; 2^1/2 = √2 is irrational.

8. x^2/3.

9. √[3]{m⁴}.

10. Rational, because √49 = 7, an integer.

11. A rational number is a value that can be written as a fraction of integers. A rational exponent is an exponent written as a fraction; it may produce either a rational or irrational value.