Topicm08a03ce090d3925a_1528449000663_0Topic

Powers with integer exponents

Levelm08a03ce090d3925a_1528449084556_0Level

Third

Core curriculumm08a03ce090d3925a_1528449076687_0Core curriculum

I. Real numbers. Basic level. The student:

1) does the operations (addition, subtraction, multiplication, division, exponentiation, extraction of roots, taking the logarithm) in the set of real numbers.

Timingm08a03ce090d3925a_1528449068082_0Timing

45 minutes

General objectivem08a03ce090d3925a_1528449523725_0General objective

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

Specific objectivesm08a03ce090d3925a_1528449552113_0Specific objectives

1. Calculating values of powers with integer exponents.

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

Learning outcomesm08a03ce090d3925a_1528450430307_0Learning outcomes

The student:

- calculates values of powers with integer exponents.

Methodsm08a03ce090d3925a_1528449534267_0Methods

1. Discussion.

2. Situational analysis.

Forms of workm08a03ce090d3925a_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm08a03ce090d3925a_1528450127855_0Introduction

The teacher introduces the subject of the lesson - calculating values of powers with integer exponents.

Procedurem08a03ce090d3925a_1528446435040_0Procedure

Task
Students revise the definition of the power with the natural exponent.

A power of number a with natural exponent (n > 1) is called a product of n elements, each equal to a. We write it down as:

Moreover, we assume that aIndeks górny 0⁰ = 1, for a ≠ 0.

[Interactive illustration]

Students analyse examples:

$6^{-1}=\frac{1}{6}$

${\left(\frac{5}{8}\right)}^{-1}=\frac{8}{5}$

${\left(0,3\right)}^{-1}={\left(\frac{3}{10}\right)}^{-1}=\frac{10}{3}$

Then together they write them down in the form of fractions: $7^{-1},{\left(\frac{1}{9}\right)}^{-1},{\left(1\frac{1}{2}\right)}^{-1},{(0,8)}^{-1}$.

They should notice that obtained results are respectively multiplicate inverses of the following numbers:
$7;\frac{1}{9};1\frac{1}{2};0,8.$

Definition of the power with exponent -1.
For any number a, a ≠ 0 we assume that aIndeks górny -1^-1 = $\frac{1}{a}.$

Using the definition and properties of operations on powers, students calculate values of power with negative exponents.m08a03ce090d3925a_1527752256679_0Using the definition and properties of operations on powers, students calculate values of power with negative exponents.

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

${\left(\frac{1}{3}\right)}^{-2}={\left({\left(\frac{{\displaystyle 1}}{{\displaystyle 3}}\right)}^{-1}\right)}^2=\frac{1}{{\displaystyle {\left(\frac{1}{3}\right)}^2}}=\frac{1}{{\displaystyle \frac{1}{9}}}=9$

Then together they define the power with an integer negative exponent.

Definition of the power with an integer negative exponent.
For each natural number n and any number a, a ≠ 0 we assume that $a^{-n}=\frac{1}{a^n}$.m08a03ce090d3925a_1527752263647_0For each natural number n and any number a, a ≠ 0 we assume that $a^{-n}=\frac{1}{a^n}$.

Using the definition students calculate values of power with integer negative exponents.

Task
Calculate.

a) $3^{-2},4^{-4},{10}^{-2},{47}^{-1},-2^{-3},{(-2)}^{-2}$

b) ${\left(\frac{1}{3}\right)}^{-3},\frac{2}{5^{-2}},{(-\frac{1}{7})}^{-1},0,8^{-2},{(-0,3)}^{-2},-{(-0,3)}^{-2}$

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

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

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

c) $-4^{-5}...{\left(-4\right)}^{-5}$

d) $1^{-4}...-{\left(-1\right)}^{-4}$

Task
Insert a proper exponent in the dotted space.

a) $\frac{2}{7}={\left(\frac{7}{2}\right)}^{...}$

b) $8={\left(\frac{1}{2}\right)}^{...}$

c) $\frac{{\displaystyle 16}}{81}={\left(\frac{3}{2}\right)}^{...}$

d) $\frac{{\displaystyle 1}}{125}=5^{...}$

Task
Calculate.

a) $1^{-5}-(-1{)}^{-5}$

b) $\frac{{\displaystyle 1}}{3}·3^2-(-3{)}^{-1}$

c) ${10}^3·0,1^{-4}$

An extra task:
Calculate the value of the expressionvalue of the expressionvalue of the expression: $-(-{(-3)}^{-2})+{(-\frac{1}{3})}^{-2}:9$.

Lesson summarym08a03ce090d3925a_1528450119332_0Lesson summary

Students do the revision exercises.

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

A power of number a with natural exponent (n > 1) is called a product of n elements, each equal to a. We write it down as:

Moreover, we assume that aIndeks górny 0⁰ = 1, for a ≠ 0.

Number a is called the base of the powerbase of the powerbase of the power and number n – the exponent of the powerexponent of the powerexponent of the power.

For any number a, a ≠ 0 we assume that aIndeks górny -1^-1 = $\frac{1}{a}.$

For each natural number n and any number a, a ≠ 0 we assume that $a^{-n}=\frac{1}{a^n}$.

Selected words and expressions used in the lesson plan

algebraic expressionsalgebraic expressionsalgebraic expressions

base of the powerbase of the powerbase of the power

exponent of the powerexponent of the powerexponent of the power

power with the integer exponentpower with the integer exponentpower with the integer exponent

value of the expressionvalue of the expressionvalue of the expression