Unit 1 Real Number System
UEQ: How do we use the properties of rational and irrational numbers and extend the properties of
exponents to rational exponents?
Lesson 2: Rational Exponents
Chapter 6 Lesson 2
Learning Goals for this Lesson:
Standards: MAFS.912.N-RN.1.1, MAFS.912.N-RN.1.2,
MAFS.912.N-RN.2.3, MAFS.912.N-Q.1.1
Students Will Know:
Students Will Be Able To:
Understand that the denominator of a rational exponent is
Define radical notation as a convention used to represent
the root index and the numerator is the exponent of the
rational exponents. Convert from radical expressions to
radicand. Extend the properties of exponents when
using rational exponents and vis versa.
justifying answers in simplified form.
Lesson Essential Question:
How can I use properties of exponents to write expressions involving radical and rational exponents?
Activating Strategy:
Have the students write and evaluate the following expressions in the simplest form. We will refer to the students
1
answers in Learning Activity 1. 1. √2 2. 0. 3̅ 3. 𝜋 4. −
5. 0
2
Key Vocabulary to Preview and Vocabulary Strategy:
Zero exponent, negative exponent, base, power, rational number, irrational number, integer, whole number,
natural number, radical symbol, root, index, nth root, square root, cube root, perfect square, perfect cube.
Lesson Instruction:
Learning Activity 1:
Introduce Vocabulary: rational numbers, irrational numbers, integer, whole numbers,
natural numbers and irrational numbers. Have the students make a venn diagram in their
cornell notes to represent the venn diagram on page 327 in their workbooks. Show and
model examples of each term.
Refer to the student’s answers to their expressions from the Activating Strategy. Review
the answers and have the students classify each answer. State whether the answers was
a rational or irrational number. If it is rational which characteristics does it have (integer,
1
whole number, natural number) 1. √2 2. 0. 3̅ 3. 𝜋 4. −
5. 0
2
Graphic Organizer:
Cornell Notes
Guided Notes: Exponents
and Radicals
Graphic Organizer: Real
Numbers Venn Diagram,
Property of Exponents
Foldable: Properties of
Exponents
Assessment Prompt for LA 1:
Quick Write: Have the students write in their cornell note books the answer to the
following question: (answer must be written in at least three complete sentences) What
is the main differences between rational and irrational numbers, and how do you
determine if a number is rational or irrational?
Differentiation:
Learning Activity 2:
Introduce Vocabulary: base, exponent, index, radical symbol, radicand, root, square root,
cube root, nth root. Show and model examples of each term.
Model and Explain to the students how to Read and Understand Radical Expressions.
Index is the positive integer in-between the “v” in the radical symbol. Radicand is a nonnegative number. Read it as the nth root of a. When the radicand it has two nth roots
one that is positive and one that is negative. For example: √25 = 52 𝑎𝑛𝑑 (−5)2
Model and Explain to students to convert radical expressions into rational exponent
forms.
Assignment:
Textbook Pages 398-403
Workbook Pages 327-336
Practice Assignment:
TB: 401-403
Basic: #32-58
Advanced: # 60-80
Extra Credit: # 81-97
1
Quick Check Homework:
34,40,44,52,52,56
1
Definition of 𝑏 𝑛 . The “b” is the base and the “n” in the fraction is the index. So 𝑏 𝑛
equals the nth root of b.
1
3
Example: 1253 means the cube root of 125. √125 3 is the index and you are finding the
root of 125. This mean what number times itself 3 times will result in a solution of 125?
Which 5 x 5 x 5 is 125. So the solution is 5
1
4
Example: √256 is equal to 2564 this means what number times itself 4 times will result
in the solution of 256. 4 x 4 x 4 x 4 = 256
See example 1 in the text book on page 398.
Assessment Prompt for LA 2:
Collaborative Pairs: Students will write the expression in either radical or rational
6
exponent form and then find its solution. Student one will use √64 and student two will
1
use 83 . The first student will explain how they got their answer to the other student and
vis versa. If a student disagrees with the other student’s answer, then they will work
together to find the correct solution to the problem.
Differentiation:
Learning Activity 3:
Model and Explain to students to convert radical expressions into rational exponent
forms.
𝑚
Definition of 𝑏 𝑛 . B is the base, n is the index but now we have another variable “m”. M
represents and functions as a power of b. So the number in the numerator is the
exponent of b and the number in the denominator is the index.
2
3
Example : 83 . 8 is the base, 3 is the index and 2 is the power. Should be written as √82 .
3
To Solve this problem √82 . Simplify the base and power. 82 = 64.
3
3
Rewrite the expression √82 as √64. What number times itself 3 times will equal 64? =4
See examples on page 399 in the text book.
Assessment Prompt for LA 3:
Think-Ink-Pair-Share: Have the students write the steps they would use to simplify
5
7296 ? The first student will explain how they got their answer to the other student and
vis versa. If a student disagrees with the other student’s answer, then they will work
together to find the correct solution to the problem.
Differentiation:
Learning Activity 4:
Model and Explain to the students how incorporate the properties of exponents in with
rational exponents.
Remind students to simplify the expressions first before they begin to try and solve them
Example: √𝑥 9 𝑦 3
Step 1: What does the question want you to do? Rewrite the expression using
3
1
information that you just learned about rational exponents. (𝑥 9 𝑦 3 )3
1
1
Step 2: Power of Products 𝑥 9×3 𝑦 3×3
Step 3: Simplify 𝑥 3 𝑦
See example 4 on page 400 in the text book.
Assessment Prompt 4:
4
Collaborative Pairs: Have student 1 simplify √𝑎4 𝑏 20 and student 2 will simplify
√𝑎𝑥 4 𝑦12 . The first student will explain how they got their answer to the other student
and vis versa. If a student disagrees with the other student’s answer, then they will work
together to find the correct solution to the problem.
4
Differentiation:
Summarizing Strategy:
1
3
Ticket Out the Door: Have the students simplify the following expression. (𝑎4 𝑏 4 )4 x √𝑏 3