Powers with natural exponents

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Potęga o wykładniku naturalnym

Learning objectives

Calculating the values of powers with natural exponents on the basis of the definition.
Determining the sign of the power.

Learning effect

Calculates the values of powers with natural exponents on the basis of the definition.
Determines the sign of the power.

Task 1

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Source: GroMar, licencja: CC BY 3.0.

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nagranie abstraktu

Find a square and a cube for each digit greater than 1. Learn them by heart.

Example 1

Analyse the following examples. Notice that in each case we need to multiply a few identical elements.

$12 \cdot 12 \cdot 12 \cdot 12 = 20736$

$\frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{64}$

$\frac{2}{3} \cdot \frac{2}{3} \cdot \frac{2}{3} \cdot \frac{2}{3} \cdot \frac{2}{3} \cdot \frac{2}{3} \cdot \frac{2}{3} = \frac{128}{2187}$

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nagranie abstraktu

Multiplying many identical elements, not just two or three, can be written down in the form of power.

Power

Definition: Power

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nagranie abstraktu

The power of number a, with a natural exponent (n > 1) is called a product of n elements, each being equal to a.

We write it down as:

a^{n} = a \cdot a \cdot a \cdot . . . \cdot a

Number a is called the base of the powerbase of the powerbase of the power, number n – the exponent of the powerexponent of the powerexponent of the power. Moreover, we assume that: $a^{0} = 1,$ for $a \neq 0$ .

Using the definition, calculate values of powers.

Task 2

Calculate.

a) $4^{2}, 3^{3}, 0^{4}, 10^{4}, 135^{0},$

b) $(- 5)^{3}, (- 3)^{0}, - 3^{3}, (- 1)^{12}, (- 4)^{2},$

c) $(\frac{1}{3})^{3}, \frac{2}{5^{2}}, (- \frac{1}{7})^{1}, (- \frac{1}{2})^{4}, \frac{- 1}{- 4^{2}} .$

Task 3

Calculate the values of powers. Think what determines the sign of the result of the exponentiation.

a) $2^{3}, 4^{2}, 6^{3},$

b) $(- 5)^{2}, (- 3)^{4}, (- 2)^{6},$

c) $(- 3)^{3}, (- 5)^{3}, (- 2)^{5} .$

Notice that:

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nagranie abstraktu

If we raise a positive number to the power with a natural exponent, we obtain a positive number.
If we raise a negative number to the power with a natural, even exponent, we obtain a positive number.
If we raise a negative number to the power with a natural, odd exponent, we obtain a negative number.

Use the information to do the tasks.

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Exercise 1

Put >, < or = in correct place.

$=$ , $>$ , $<$ , $<$ , $=$ , $>$ , $<$ , $>$ , $=$

$3^{2}$ ............ $2^{3}$
$1^{2}$ ............ $(- 1)^{2}$
$- 3^{3}$ ............ $3^{3}$
$4^{3}$ ............ $(- 4)^{3}$
$- 1^{5}$ ............ $(- 1)^{5}$
$- 3^{2}$ ............ $- (- 3)^{2}$

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Exercise 2

Arrange the numbers from the smallest one to the greatest one.

$2^{1}$
$24^{0}$
$(- 2, 4)^{2}$
$(\frac{1}{2})^{4}$
$2^{3}$
$(- 16)^{1}$
$(- 2)^{1}$
$(- 2)^{2}$

Task 4

Your task is to match the powers and their properties in pairs.

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Rysunek przedstawia umieszczone w różnych częściach ekranu prostokąt i kwadraty. Wewnątrz figur znajdują się potęgi liczby 2 - dwa do pierwszej, dwa do drugiej ….dwa do dziesiątej oraz liczby: dwa, cztery , osiem, szesnaście, trzydzieści dwa, sześćdziesiąt cztery, sto dwadzieścia osiem, dwieście pięćdziesiąt sześć, pięćset dwanaście, tysiąc dwadzieścia cztery. — Source: GroMar, licencja: CC BY 3.0.

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Task 5

An extra task:

Without performing the actual calculations, determine if the result is a positive or negative number.

a) $(- \frac{2}{7})^{4} \cdot (- 0, 8)^{5}$

b) $- {(- 3, 4)}^{7} \cdot {\frac{(- 1)}{- 3}}^{3}$

c) $\frac{- 1^{2} \cdot (- {(- 6)}^{3})}{- {(- 0, 4)}^{3} \cdot {(- 2)}^{5}}$

Do the revision exercises.

Exercises

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Exercise 3

Wersja alternatywna ćwiczenia: Determine which sentence is true. Możliwe odpowiedzi: 1. The sum of the numbers

7^{3}

and

(- 7)^{3}

is a positive number., 2. The sum of the numbers

5^{2}

and

{(-5)}^{2}

is a positive number., 3. The cube of any number is always greater than the square of the same number., 4. The cube of any number greater than 1 is always greater than the square of the same number., 5. The numbers 5 and -5 raised to the same power are equal., 6. The numbers 3 and -3 raised to the same even power are equal.

Determine which sentence is true.

The sum of the numbers $7^{3}$ and $(- 7)^{3}$ is a positive number.
The sum of the numbers $5^{2}$ and ${(-5)}^{2}$ is a positive number.
The cube of any number is always greater than the square of the same number.
The cube of any number greater than 1 is always greater than the square of the same number.
The numbers 5 and -5 raised to the same power are equal.
The numbers 3 and -3 raised to the same even power are equal.

zadanie

Source: GroMar, licencja: CC BY 3.0.

Exercise 4

Calculate.

a) $- 2^{2}$

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

c) ${(- 2)}^{2}$

d) $- 0, 2^{2}$

e) ${(- 0, 2)}^{2}$

f) ${(- \frac{1}{2})}^{2}$

a) $- 4$

b) $- \frac{1}{4}$

c) $4$

d) $- 0, 04$

e) $0, 04$

f) $\frac{1}{4}$

Exercise 5

Calculate the consecutive 11 powers of number 2, starting with the exponent 0. Describe the solution in English.

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Exercise 6

Indicate which pairs of expressions or words are translated correctly.

podstawa potęgi - the base of the power
wykładnik potęgi - the exponent of the power
potęga o wykładniku naturalnym - power with the natural exponent
kwadrat liczby - the cube of the number
sześcian liczby - the square of the number

zadanie

Source: GroMar, licencja: CC BY 3.0.

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Match Polish terms with their English equivalents.

wykładnik potęgi
the base of the power
the exponent of the power
power with the natural exponent
the cube of the number
the square of the number
sześcian liczby
podstawa potęgi
potęga o wykładniku naturalnym
kwadrat liczby

Source: Zespół autorski Politechniki Łódzkiej, licencja: CC BY 3.0.

Glossary

base of the power

podstawa potęgi

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wymowa w języku angielskim: base of the power

exponent of the power

wykładnik potęgi

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Source: GroMar, licencja: CC BY 3.0.

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wymowa w języku angielskim: exponent of the power

power with the natural exponent

potęga o wykładniku naturalnym

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Source: GroMar, licencja: CC BY 3.0.

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Source: GroMar, licencja: CC BY 3.0.

wymowa w języku angielskim: power with the natural exponent

square of the number

kwadrat liczby

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Source: GroMar, licencja: CC BY 3.0.

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wymowa w języku angielskim: square of the number

cube of the number

sześcian liczby

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Source: GroMar, licencja: CC BY 3.0.

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wymowa w języku angielskim: cube of the number

Keywords

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

square of the numbersquare of the number square of the number

cube of the numbercube of the number cube of the number

Scenariusz

Exercises

Glossary

Keywords