Topicm9fc7485873b96670_1528449000663_0Topic

The power of the rational exponentexponentexponent

Levelm9fc7485873b96670_1528449084556_0Level

Third

Core curriculumm9fc7485873b96670_1528449076687_0Core curriculum

I. Real numbers. The student:

1) performs operations (addition, subtraction, multiplication, division, exponentiation, square roots, logarithms) in a set of real numbers;

4) uses the relationship of roots with exponentiation and the rules of operations on powers and roots.

Timingm9fc7485873b96670_1528449068082_0Timing

45 minutes

General objectivem9fc7485873b96670_1528449523725_0General objective

Interpreting and manipulating information presented in both mathematical and popular science texts, as well as in the form of graphs, diagrams, tables.

Specific objectivesm9fc7485873b96670_1528449552113_0Specific objectives

1. Communicating in English, developing mathematical, scientific, technical and IT competences; developing learning skills.

2. Learning the definition of the power with rational exponentexponentexponent.

3. Getting to know the operations on powers with rational exponents.

Learning outcomesm9fc7485873b96670_1528450430307_0Learning outcomes

The student:

- knows the definition of the power with rational exponentexponentexponent,

- knows the operations on powers with rational exponents.

Methodsm9fc7485873b96670_1528449534267_0Methods

1. Diamond ranking.

2. Competition.

Forms of workm9fc7485873b96670_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm9fc7485873b96670_1528450127855_0Introduction

Students, working in groups with the method of diamond rankings review their knowledge of exponentiation. The collected information is placed on the boards. After finishing the task, they present their boards.

The teacher verifies the information and explains the doubts.

Procedurem9fc7485873b96670_1528446435040_0Procedure

The teacher informs students that the aim of the lesson is to learn the definitions and properties of powers with rational exponents.

Definition

- For any non‑negative number a and a natural number n greater than 1 we accept:

- For a natural number n greater than 1, an integer m and a positive number a we accept:

Task
Students, working in pairs, analyze the material presented in the Interactive illustration. They make hypotheses and check them. They formulate the rules of powers with a rational exponentexponentexponent.

[Interactive illustration]

Definition

- For any number a > 0, the natural number n > 1 and the integer m, we assume:

Conclusion:

- The rules of the powers with a rational exponent are the same as the rules of operations on powers with a whole exponent.m9fc7485873b96670_1527752263647_0- The rules of the powers with a rational exponent are the same as the rules of operations on powers with a whole exponent.

Students, working alone, take part in a task competition.

The competition has three stages. For each correctly performed calculation, the student receives 1 point, for the mistake (-1). Students who have obtained a minimum of 3 points pass to the next stage.

I stage.

Task
Calculate.

a) ${64}^{\frac{1}{3}}$

b) ${243}^{-\frac{1}{3}}$

c) $(\frac{81}{625}{)}^{-0,75}$

d) $(\frac{27}{8}{)}^{-\frac{4}{3}}$

Task
Calculate.

a) $2·{16}^{-1,5}·{32}^{1,2}$

b) $(3^{\frac{1}{3}}·{27}^{\frac{2}{3}}·81{)}^{-0,75}$

c) $4^{-4}·{64}^{\frac{2}{3}}·{256}^{\frac{5}{4}}$

II Stage.

Task
Write the number in the form of one power with a rational exponentexponentexponent.

a) $3\sqrt{3\sqrt{3}}$

b) $5\sqrt{25\sqrt[3]{125}}$

c) $81·\sqrt[3]{9\sqrt{9}}$

d) $16\sqrt[3]{4\sqrt{256}}$

Task
Calculate.

a) ${72}^{\frac{1}{2}}·2^{\frac{1}{2}}+{12}^{\frac{1}{3}}·{12}^{\frac{2}{3}}$

b) ${32}^{\frac{1}{3}}·2^{\frac{1}{3}}+6^{\frac{1}{3}}·{36}^{\frac{1}{3}}$

III stage.

Task
Calculate the expressions.

a) $3·0,3^{-1}+4·8^{\frac{2}{3}}-12·{27}^{-\frac{1}{3}}$

b) $(0,{125}^{-\frac{2}{3}}·0,{25}^2)+({81}^{0,5}·9^{-2}{)}^{-0,25}$

The teacher summarizes the results of the competition. The students who scored the most points are rewarded with grades.

Students solve the task on their own using the information learned.

Task
Simplify your expression by applying the rules of operations on the powers. Provide the necessary assumptions.

a) $x^{0,5}·x^{-1,25}$

b) $x^{\frac{2}{3}}:{\left(\sqrt[3]{x}\right)}^5$

c) $\sqrt{x}·x^{0,75}:x^{-0,5}$

An extra task:
Prove, that the number ${(7+\sqrt{24})}^{0,5}-{(7-\sqrt{24})}^{0,5}$ is a whole number.

Lesson summarym9fc7485873b96670_1528450119332_0Lesson summary

Students do the revision exercises and formulate conclusions to remember.

- For any number a > 0 the natural number n > 1 and the integer m, we assume:

- The rules of operations on powers with a rational exponentexponentexponent are the same as the rules of operations on powers with a whole exponent.

Selected words and expressions used in the lesson plan

division of powers with the same basedivision of powers with the same basedivision of powers with the same base

division of powers with the same exponentdivision of powers with the same exponentdivision of powers with the same exponent

exponentexponentexponent

exponentationexponentationexponentation

multiplication of powers with the same basemultiplication of powers with the same basemultiplication of powers with the same base

multiplication of powers with the same exponentmultiplication of powers with the same exponentmultiplication of powers with the same exponent

power of the powerpower of the powerpower of the power

power with a rational exponentpower with a rational exponentpower with a rational exponent