Topicme8f34b6a2dd525d6_1528449000663_0Topic

The product and the quotient of powers of the same exponents

Levelme8f34b6a2dd525d6_1528449084556_0Level

Second

Core curriculumme8f34b6a2dd525d6_1528449076687_0Core curriculum

I. Powers of rational bases. The student:

3) multiplies powers of different bases and equal exponents.

Timingme8f34b6a2dd525d6_1528449068082_0Timing

45 minutes

General objectiveme8f34b6a2dd525d6_1528449523725_0General objective

Using mathematical objects, interpreting mathematical concepts.

Specific objectivesme8f34b6a2dd525d6_1528449552113_0Specific objectives

1. Applying the theorem about the product and the quotient of powers of the same exponentsquotient of powers of the same exponentsquotient of powers of the same exponents.

2. Calculating values of powers using the theorem about the product and the quotient of powers of the same exponents.

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

Learning outcomesme8f34b6a2dd525d6_1528450430307_0Learning outcomes

The student:

- applies the theorem about the product and the quotient of powers of the same exponentsquotient of powers of the same exponentsquotient of powers of the same exponents,

- calculates values of powers using the theorem about the product and the quotient of powers of the same exponents.

Methodsme8f34b6a2dd525d6_1528449534267_0Methods

1. Discussion.

2. Situational analysis.

Forms of workme8f34b6a2dd525d6_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionme8f34b6a2dd525d6_1528450127855_0Introduction

Students remember that exponentiation is a shorter version of multiplication. They give proper examples.

Procedureme8f34b6a2dd525d6_1528446435040_0Procedure

Students analyse the examples.

$3^3\cdot 4^3=(3\cdot 3\cdot 3)\cdot (4\cdot 4\cdot 4)=3\cdot 3\cdot 3\cdot 4\cdot 4\cdot 4=(3\cdot 4)\cdot (3\cdot 4)\cdot (3\cdot 4)={(3\cdot 4)}^3$

$2^5\cdot 5^2\cdot 7^2=(2\cdot 2)\cdot (5\cdot 5)\cdot (7\cdot 7)=2\cdot 5\cdot 7\cdot 2\cdot 5\cdot 7=(2\cdot 5\cdot 7)\cdot (2\cdot 5\cdot 7)={(2\cdot 5\cdot 7)}^2$

Together they wonder what is the relations between bases of the multiplied powers of the same exponents and the base and exponent of the product. They consider the problem based on own examples.

Task
They check their assumptions using computers and draw a conclusion.

[Slideshow]

Conclusion:

- The product of powers of the same exponents is equal to the power of the same exponent and the base equal to the product of the multiplied powers’ bases
$a^n·b^n={\left(a·b\right)}^n$
where:
n - is a natural number different than 0.me8f34b6a2dd525d6_1527752256679_0- The product of powers of the same exponents is equal to the power of the same exponent and the base equal to the product of the multiplied powers’ bases
$a^n·b^n={\left(a·b\right)}^n$
where:
n - is a natural number different than 0.

Discussion – what formula will be obtained if we divide powers of the same exponent?

Students write down proper examples. Using the division of fractions, they write down division in the form of multiplication, where one of the elements is the reverse of the divisor.

Students should draw the following conclusion:

- The quotient of powers of the same exponentsquotient of powers of the same exponentsquotient of powers of the same exponents is equal to the power of the same exponent and base equal to the quotient of the bases of the dividend and the divisor $\frac{a^n}{b^n}={\left(\frac{a}{b}\right)}^n$ for $b\ne 0$ and natural number n greater than 0.

Students are divided into groups. The teacher asks questions one group by one, respectively from tasks 1, 2, 3, 4 and 5. Time to do the example: 1 min. The group that answers correctly all questions within the time limit, gets three pluses or a grade.

Task
Calculate.

a) $2^4·{\left(\frac{1}{4}\right)}^4$

b) $(-3{)}^3·{\left(-\frac{1}{27}\right)}^3$

c) ${\left(\frac{{\displaystyle 1}}{2}\right)}^2·{\left(-8\right)}^2$

d) ${\left(-0,01\right)}^2·{10}^2$

Task
Write without parentheses.

a) ${\left(4x\right)}^2$

b) ${\left(\frac{{\displaystyle 1}}{3}xy\right)}^4$

c) ${\left(0,2x^2\right)}^2$

d) ${\left(3x^3·\frac{1}{4}y\right)}^2$

Task
Calculate.

a) $4^3:(-4{)}^3$

b) ${\left(\frac{1}{3}\right)}^1:9^1$

c) $0,1^5:0,2^5$

d) ${\left(-\frac{1}{4}\right)}^2:{\left(-\frac{{\displaystyle 1}}{{\displaystyle 16}}\right)}^2$

Task
Write without parentheses.

a) ${\left(2x:y\right)}^3$

b) ${\left(3x:y\right)}^4$

c) ${\left(0,2:x^2\right)}^2$

d) ${\left(5x^2:4y\right)}^3$

Task
Calculate.

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

b) ${\left(3a^2{b}^2:3a\right)}^2$

c) ${\left(3a:b^3\right)}^3$

d) ${\left(\left(9a^{3}b\right):\left(a{b}^2\right)\right)}^2$

An extra task:
Fill the base of the powerbase of the powerbase of the power.

${(0,125)}^2:{\left(\frac{1}{8}\right)}^2:2^2=..{.}^2$

Lesson summaryme8f34b6a2dd525d6_1528450119332_0Lesson summary

Students do the revision exercises.

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

- The product of powers of the same exponentsproduct of powers of the same exponentsproduct of powers of the same exponents is equal to the power of the same exponent and the base equal to the product of the multiplied powers’ bases
$a^n·b^n={\left(a·b\right)}^n$
where:
n - is a natural number different than 0.

- The quotient of powers of the same exponents is equal to the power of the same exponent and base equal to the quotient of the bases of the dividend and the divisor $\frac{{\displaystyle a^n}}{b^n}={\left(\frac{a}{b}\right)}^n$ for $b\ne 0$ and natural number n greater than 0.me8f34b6a2dd525d6_1527752263647_0- The quotient of powers of the same exponents is equal to the power of the same exponent and base equal to the quotient of the bases of the dividend and the divisor $\frac{{\displaystyle a^n}}{b^n}={\left(\frac{a}{b}\right)}^n$ for $b\ne 0$ and natural number n greater than 0.

Selected words and expressions used in the lesson plan

base of the powerbase of the powerbase of the power

exponent of the powerexponent of the powerexponent of the power

power with the natural exponentpower with the natural exponentpower with the natural exponent

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