Topicm1ebb89c67a5e5272_1528449000663_0Topic

Operations on powers with integer exponents

Levelm1ebb89c67a5e5272_1528449084556_0Level

Third

Core curriculumm1ebb89c67a5e5272_1528449076687_0Core curriculum

I. Real numbers. The student:

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

Timingm1ebb89c67a5e5272_1528449068082_0Timing

45 minutes

General objectivem1ebb89c67a5e5272_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 objectivesm1ebb89c67a5e5272_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 outcomesm1ebb89c67a5e5272_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.

Methodsm1ebb89c67a5e5272_1528449534267_0Methods

1. Discussion.

2. Thematical contest.

Forms of workm1ebb89c67a5e5272_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm1ebb89c67a5e5272_1528450127855_0Introduction

The teacher introduces the subject of the lesson – applying rules of operations on powersoperations on powersoperations on powers with integer exponents.

Procedurem1ebb89c67a5e5272_1528446435040_0Procedure

Task
Using the computer, students get to know the theorem about dividing powers of the same exponents.m1ebb89c67a5e5272_1527752256679_0Using the computer, students get to know the theorem about dividing powers of the same exponents.

[Slideshow]

In groups, students formulate the theorem about operations on powers with integer exponents, based on proper formulas about operations on powers with natural exponents. The first group that writes correctly all formulas gets pluses.m1ebb89c67a5e5272_1527752263647_0formulate the theorem about operations on powers with integer exponents, based on proper formulas about operations on powers with natural exponents. The first group that writes correctly all formulas gets pluses.

Theorem

- If a and b are numbers different than zero, x is an integer number, then:

a) $a^x·b^x=(a·b{)}^x$

b) $\frac{a^x}{b^x}={\left(\frac{a}{b}\right)}^x$

Students are divided into groups. The teacher gives each group one question, respectively from tasks 1, 2, 3 and 4.

Time for answering one question is 1 min. The group that answers all the questions correctly and within the time limit, gets three pluses or a grade.

Task
Calculate.

a) $2^{-3}·{\left(\frac{1}{8}\right)}^{-3}$

b) $3^{-2}·{\left(\frac{1}{9}\right)}^{-2}$

c) ${\left(-5\right)}^{-1}·{\left(-\frac{1}{25}\right)}^{-1}$

d) ${\left(-0,01\right)}^{-5}·{\left(10\right)}^{-5}$

e) $3^{-3}:{\left(-3\right)}^{-3}$

Task
Fill in bases of powers.

a) $2^{-3}·4^{-3}=..{.}^3$

b) ${\left(\frac{1}{3}\right)}^{-4}·9^{-4}=..{.}^4$

c) ${\left(2\frac{1}{2}\right)}^{-5}·{\left(-3\right)}^{-5}=..{.}^5$

d) ${\left(-\frac{5}{4}\right)}^{-4}·{\left(-\frac{4}{5}\right)}^{-5}=..{.}^4$

e) ${30}^{-3}:{15}^{-3}=..{.}^3$

Task
Assuming that $a\ne 0$ write the expression in the form of one power.

a) $(a^{12}·a^{-3}):(a^{-4}·a)$

b) $a^6:\left[a^5:\left(a^{-7}·a^{-1}\right)\right]$

c) $a^8·\left[a^{-3}:\left(a^{-7}·a^{-1}\right)\right]$

d) $\frac{a^{-3}·a^2:a^{-2}}{a^{-4}:a^6}$

e) $\frac{a^5·\left[\left(a^{-3}·a^5\right):\left(a^{-4}:a^3\right)\right]}{a^6:\left(a^{-3}·a\right)}$

Task
Insert a proper sign in the dotted space <, >, =.

a) $1,2^{-3}...1,1^{-3}$

b) ${\left(\frac{1}{5}\right)}^{-6}...{\left(\frac{1}{4}\right)}^{-6}$

c) ${\left(0,75\right)}^{-5}...{\left(\frac{3}{4}\right)}^{-5}$

d) $2^{-3}...{\left(2^3\right)}^{-3}$

e) ${\left(3^{-1}\right)}^4...3^{-7}$

An extra task:
Calculate

$\frac{4^3-2·4^2}{4^{-3}}$.

Lesson summarym1ebb89c67a5e5272_1528450119332_0Lesson summary

Students do the revision exercises.

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

- If a and b are numbers different than zero, x is an integer number, then:

a) $a^x·b^x=(a·b{)}^x$

b) $\frac{a^x}{b^x}={\left(\frac{a}{b}\right)}^x$

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 exponentsproduct of powers of the same exponentsproduct of powers of the same exponents

quotient of powers of the same exponentsquotient of powers of the same exponentsquotient of powers of the same exponents