Topicm27c5572af306df8e_1528449000663_0Topic

Calculating powerscalculating powersCalculating powers with integer exponents

Levelm27c5572af306df8e_1528449084556_0Level

Third

Core curriculumm27c5572af306df8e_1528449076687_0Core curriculum

I. Real numbers. The student:

4) uses the corelation between roots and powers and rules of operations on powersoperations on powersoperations on powers and roots.

Timingm27c5572af306df8e_1528449068082_0Timing

45 minutes

General objectivem27c5572af306df8e_1528449523725_0General objective

Doing calculations on real numbers, also using the calculator, applying mathematical laws of operation during transforming algebraic expressions and using this abilities in theoretical and realistic problems.

Specific objectivesm27c5572af306df8e_1528449552113_0Specific objectives

1. Applying theorems about the product and the quotient of powers with the same bases and the theorem about the exponentiation of powersexponentiation of powersexponentiation of powers.

2. Calculating values of powers while using theorems about the product and the quotient of powers with the same bases and the theorem about the exponentiation of powers.

3. Communicating in English, developing basic mathematical, computer and scientific competences, developing learning skills.

Learning outcomesm27c5572af306df8e_1528450430307_0Learning outcomes

The student:

- applies theorems about the product and the quotient of powers with the same bases and the theorem about the exponentiation of powers,

- calculates values of powers while using theorems about the product and the quotient of powers with the same bases and the theorem about the exponentiation of powersexponentiation of powersexponentiation of powers.

Methodsm27c5572af306df8e_1528449534267_0Methods

1. Discussion.

2. Situational analysis.

Forms of workm27c5572af306df8e_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm27c5572af306df8e_1528450127855_0Introduction

At home students revise theorems about the product and the quotient of powers with the same bases and the theorem about the exponentiation of powersexponentiation of powersexponentiation of powers.

Procedurem27c5572af306df8e_1528446435040_0Procedure

Task
Using the computer, students get to know the theorem about dividing powers of the same bases.m27c5572af306df8e_1527752263647_0Using the computer, students get to know the theorem about dividing powers of the same bases.

[Slideshow]

Based on the proper theorem about operations on powers with natural exponents, students formulate a theorem about operations on powers with integer exponents.m27c5572af306df8e_1527752256679_0Based on the proper theorem about operations on powers with natural exponents, students formulate a theorem about operations on powers with integer exponents.

Theorem

If a is a number different that 0; x, y - are integers then:

- $a^x·a^y=a^{x+y}$

- $\frac{a^x}{a^y}=a^{x-y}$

- ${\left(a^x\right)}^y=a^{x·y}$

Students solve tasks in 4‑5 people groups.

Task
Calculate.

a) $2^{-3}·2^{-1}$

b) $3^{-3}·3^0$

c) ${\left(\frac{1}{2}\right)}^{-7}·{\left(\frac{{\displaystyle 1}}{{\displaystyle 2}}\right)}^7$

d) $2^{-3}:2^{-2}$

e) $(-1{)}^{-4}:(-1{)}^2$

f) $3^{-2}:3^0$

Task
Write the exponent of the powerexponent of the powerexponent of the power.

a) $2^2·2^{-4}=2^{...}$

b) $3^{18}·3·3^{-4}=3^{...}$

c) $2^5:2^{-4}=2^{...}$

d) ${\left(-5\right)}^{-3}:{\left(-5\right)}^3:-{\left(5\right)}^5={\left(-5\right)}^{...}$

e) $4^4·4^{-7}:4^{-3}=4^{...}$

Task
Write in the form of one power.

a) ${\left(9^{-2}\right)}^5$

b) ${\left(3^3\right)}^{-4}$

c) ${\left({\left(\frac{3}{5}\right)}^{-5}\right)}^{-5}$

d) ${\left({\left(-6\right)}^3\right)}^{-4}$

Task
Match equal expressions.

a) $a^{-6}·a^5$

b) $a^{-4}·a^{-1}$

c) $a^8:a^5·a^{-2}$

d) $a^2:a^{-2}$

1. $a^4$

2. $a^{-1}·a·a$

3. $a^{-7}:a^{-2}$

4. $a^6:a^{-7}$

The group chosen by the teacher presents solutions. The teacher grades the groups’ work. If the group also did the extra task, it gets the highest mark.

An extra task:
Write in the form of one power

$5^{-4}+5^{-4}+5+5^{-4}+5^{-4}$.

Lesson summarym27c5572af306df8e_1528450119332_0Lesson summary

Students do the revision exercises.

Then together they sum‑up the classes, by formulating the conclusions to memorise.

If a is a number different that 0; x, y - are integers then:

- $a^x·a^y=a^{x+y}$

- $\frac{a^x}{a^y}=a^{x-y}$

- ${\left(a^x\right)}^y=a^{x·y}$

Selected words and expressions used in the lesson plan

base of the powerbase of the powerbase of the power

calculating powerscalculating powerscalculating powers

exponent of the powerexponent of the powerexponent of the power

exponentiation of powersexponentiation of powersexponentiation of powers

operations on powersoperations on powersoperations on powers

product of powers of the same basesproduct of powers of the same basesproduct of powers of the same bases

quotient of powers of the same basesquotient of powers of the same basesquotient of powers of the same bases