Topicm1aa6f9085e6ed5eb_1528449000663_0Topic

Operations on powers with natural exponents

Levelm1aa6f9085e6ed5eb_1528449084556_0Level

Second

Core curriculumm1aa6f9085e6ed5eb_1528449076687_0Core curriculum

I. Powers with rational bases. The student:

1) writes the product of equal elements in the form of a power with integer, positive exponents;

2) multiplies and divides powers with integer, positive exponents;

3) multiplies and divides powers of different bases and equal exponents;

4) raises a power to a power.

Timingm1aa6f9085e6ed5eb_1528449068082_0Timing

45 minutes

General objectivem1aa6f9085e6ed5eb_1528449523725_0General objective

Using mathematical objects, interpreting mathematical concepts.

Specific objectivesm1aa6f9085e6ed5eb_1528449552113_0Specific objectives

1. Using mathematical objects, interpreting mathematical concepts.

2. Transforming algebraic expressions containing powers.

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

Learning outcomesm1aa6f9085e6ed5eb_1528450430307_0Learning outcomes

The student:

- performs operations on powersoperations on powersoperations on powers with natural exponents,

- transforms algebraic expressions containing powers.

Methodsm1aa6f9085e6ed5eb_1528449534267_0Methods

1. Flipped clasroom.

2. Group work.

Forms of workm1aa6f9085e6ed5eb_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm1aa6f9085e6ed5eb_1528450127855_0Introduction

Before the class, students should search for ways of determining the last digit of powers of one‑digit numbers of natural exponents.

The class start with revision of the order of operations on algebraic expressions.

Procedurem1aa6f9085e6ed5eb_1528446435040_0Procedure

Task
Students use computers to revise the way of dividing powers with natural exponents.

[Slideshow]

Students in groups write down all formulas of theorems about operations of powers with natural exponents. The group that writes all the formulas correctly first get pluses.

Definition - formulas about operations on powersoperations on powersoperations on powers.

If a is a number different than zero, n is a natural positive number, then:

- $a^n\cdot a^m=a^{n+m},$

- $\frac{a^n}{a^m}=a^{n-m},n>m,$

- ${\left(a^n\right)}^m=a^{n\cdot m},$

- $a^n\cdot b^n={(a\cdot b)}^n,$

- $\frac{a^n}{b^n}={\left(\frac{a}{b}\right)}^n.$

One of the groups presents the method of calculating the last digit of the number $2^n$ 
($n$ – natural positive number).

Based on this methods, students do the exercise.

Task
Give the last digit of the number.

a) $2^{24}$

b) $3^{21}$

c) $5^{43}$

Task
Calculating using the theorem about the product of powers of the same exponents.m1aa6f9085e6ed5eb_1527752256679_0Calculating using the theorem about the product of powers of the same exponents.

a) $1,5^4·2^5$

b) $0,1^3·{10}^4$

c) ${\left(\frac{1}{5}\right)}^2·{25}^3$

Task
Calculate the numerical value of the expression.

a) ${\left(2a^2\right)}^3:{\left(\frac{1}{4}{a}^2\right)}^2$ for $a=-2$

b) ${\left(4x^0{y}^4\right)}^5:{\left(2x^{3}y\right)}^4$ for $x=-1$ and $y=-\frac{1}{2}$

c) ${\left(\frac{4a^2}{5b}\right)}^3·{\left(\frac{5b^2}{a}\right)}^3$ for $a=-\frac{1}{3}$ and $b=-1$

Task
Write the power $2^{\mathrm{25}}$ in the form of two product of powers.
a) Of the same exponents.
b) Of the same bases.m1aa6f9085e6ed5eb_1527752263647_0Write the power $2^{\mathrm{25}}$ in the form of two product of powers.
a) Of the same exponents.
b) Of the same bases.

Task
Fill in the base of the powerbase of the powerbase of the power.

a) ${\left(3x\right)}^2\cdot x^2=..{.}^2$

b) ${\left(y^3\right)}^4\cdot {\left(y\right)}^3=..{.}^3$

c) ${\left(2x\right)}^3\cdot {\left(5x\right)}^3=..{.}^3$

d) ${\left(y^2\right)}^5\cdot {\left(y\right)}^5=..{.}^5$

The group chosen by the teacher presents the solutions. The teacher evaluates the group’s work. A group that also solves the extra task, get the highest mark.

An extra task:
Prove that number $2^{12}+2^{11}+2^9$ can be divided by 13.

Lesson summarym1aa6f9085e6ed5eb_1528450119332_0Lesson summary

Students do the revision exercises.

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

Formulas about operations of powers of natural, positive exponents and bases different than zero:

- $a^n\cdot a^m=a^{n+m},$

- $\frac{a^n}{a^m}=a^{n-m},n>m,$

- ${\left(a^n\right)}^m=a^{n\cdot m},$

- $a^n\cdot b^n={(a\cdot b)}^n,$

- $\frac{a^n}{b^n}={\left(\frac{a}{b}\right)}^n.$

Selected words and expressions used in the lesson plan

base of the powerbase of the powerbase of the power

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

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

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