Unit 2: POWERS AND ROOTS

Unit 2: POWERS AND ROOTS

2.1.- POWERS

Example:

Susan is going to buy six boxes of chocolates. There are six rows in each box and six chocolates in each row. Each chocolate costs 0.06 €. How much do the six boxes cost in total?

There will be 6 6 6 216× × = _{chocolates in total.}

Since each chocolate costs 0.06 €, the six boxes will cost 216 0.06 12.96 €× =

Repeated multiplications such as 6 · 6 · 6 can be written in index notation as 63_.

You read 63 as ‘six to the power of 3’ or ‘six cubed’.

You use powers when factorising, for example: 3

24 2 2 2 3 2 3= ⋅ ⋅ ⋅ = ⋅

In a power, the number or expression used as a factor is called base.

And the number that indicates how many times the base is used as factor is called exponent (or index).

Exercise 1:

Complete the table below:

Power Base Exponent

6

2

5 3

4

a

m 5

Exercise 2:

Write the first twenty square numbers.

2 2 2 2 16 = ⋅ ⋅ ⋅ = 4

2

Exercise 3:

Find out the value of the bases and the exponents in each expression.

a) a4 =16 g) 2x =64

b) a2 =25 h) 3y =81

c) a3 =64 i) 6z =36

d) a4 =2401 _{j) 8}m =₅₁₂

e) a3 =1000 k) 10n =10 000 f) a10 =1024 _{l) 30}t =_{810 000}

Exercise 4:

Write the tree following terms in this sequence:

0 1 8 27 64 ? ? ?

Exercise 5:

Express as a power of base 4 and calculate the number of windows that there are in these buildings.

2.2.- POWERS OF TEN

Look at the following powers of ten:

2

10 =10 10 100⋅ =

3

10 =10 10 10 1 000⋅ ⋅ =

4

10 =10 10 10 10 10 000⋅ ⋅ ⋅ =

5

10 =10 10 10 10 10 100 000⋅ ⋅ ⋅ ⋅ =

9

10 =10 10 ... 10 1 000 000 000⋅ ⋅ ⋅ =

Exercise 6:

Express as powers of base ten.

a) A thousand b) A million c) A thousand million d) A billion

Expressing numbers in expanded form

Remember that in our decimal number system the value of each digit depends on its place in the number. Each place is 10 times the value of the next place to its right. Therefore, the decimal system is based upon powers of ten.

H un d re d M ill io ns T e n M ill io ns M ill io ns H un d re d T h ou sa nd s T e n T h ou sa nd s T h ou sa nd s H un d re d s T e ns O ne s 1 0 0 ,0 0 0 ,0 0 0 1 0 ,0 0 0 ,0 0 0 1 ,0 0 0 ,0 0 0 1 0 0 ,0 0 0 1 0 ,0 0 0 1 ,0 0 0 1 0 0 1 0 ₁ 8

10 ₁₀7 ₁₀6 ₁₀5 ₁₀4 ₁₀3 ₁₀2 ₁₀1 ₁₀0

Examples:

741 the 4 in 741 is in the

tens

place. Its value is 4

tens

, or 40.

7415 the 4 in 7415 is in the hundreds place. Its value is 4

hundreds

, or 400.

Therefore you can use powers of ten to express a number.

Example:

3 2 1 0

8, 435 8, 000 400 30 5 8 10= + + + = ⋅ +4 10⋅ +3 10⋅ +5 10⋅

The number is expressed as a sum of the values of each digit. This way to write a number is called expanded form.

Exercise 7:

Complete the table below:

Number Expanded form

28,563 3,428,567

4 3 2

5 10⋅ +6 10⋅ +8 10⋅ +5 10 5⋅ +

40,500,080

7 6 4 2

4 10⋅ +9 10⋅ +5 10⋅ +2 10⋅

9 8 2 7

Large numbers and powers of ten

You can use powers of ten to write large numbers.

Example:

A light-year is a unit of distance. It is the distance that light travels in one year.

Let’s calculate that distance:

The speed of light is 300 000 kilometres each second (km/s).

Since there are 31 536 000 seconds in one year, the distance that light travels in one year will be 300 000 · 31 536 000 = 9 460 800 000 000 kilometres.

You can round this number to 2 significant figures

Exercise 8:

In our solar system, we tend to describe distances in terms of the Astronomical Unit (AU). The AU is defined as the average distance between the Earth and the Sun. It is approximately 150 million km. Write this number as a multiple of powers of ten.

Exercise 9:

Mercury is about 1/3 of an AU from the Sun and Pluto averages about 40 AU from the Sun. How many kilometres are there between Mercury and the Sun? How many kilometres are there between Pluto and the Sun? Write both numbers as multiples of powers of ten.

9 460 800 000 000

11 95 10⋅

9 500 000 000 000

Exercise 10

Write the following numbers as multiples of powers of ten

a) The distance to the next nearest big galaxy, the Andromeda Galaxy, is 21 000 000 000 000 000 000 km.

b) The number of red blood cell in a human being is 25 000 000 000.

c) The number of atoms in a silver gram is 5 600 000 000 000 000 000 000.

Exercise 11

Write the value of x in each case:

a) x

52936428 53 10≈ ⋅ _{b) 19270 000 000 000 19 10}≈ ⋅ x

2.2.- PROPERTIES OF POWERS

Powers of the same number can be multiplied and divided.

3 4 7

5 5⋅ =(5 5 5) (5 5 5 5) 5⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = 3 4

5 +

7 5 =

6

2

3 3 3 3 3 3

3

⋅ ⋅ ⋅ ⋅

= ⋅3

3 3⋅

4

3 3 3 3 3

= ⋅ ⋅ ⋅ =

6 2 3 −

4 3 =

( )

2 3 2 2 2 2 2 2 6

5 5 5 5 5 + + 5

= ⋅ ⋅ = =

2 3 5 ⋅

6 5 =

Simplified, the index laws are:

Product of powers of Division of powers of

the same base the same base Power of a power DID YOU KNOW?

In 2006, astronomers decided that Pluto could not be regarded as a planet, due to its erratic orbit and small size. There are now considered to be only 8 planets in our solar system.

When multiplying, you add the indices.

When dividing, you subtract the indices.

And when finding a ‘power of a power’, you multiply the indices.

m n m n

a

a

a

+

⋅

=

_{a : a}

m n

_a

m n−

=

( )

a

m n

a

m n⋅

Power of a product Power of a division

Any number (except 0) raised to the power of zero is equal to 1.

Proof:

n 0 n n

n a

a a 1

a

−

= = =

Examples: 0

7 =1 ₁₀0 =₁

(00 is not defined, it doesn’t actually means anything)

Any number to the power of 1 is just the number itself.

Exercise 12

Compute in a simple way.

a) 4 4

5 2⋅ _{b) 4 5}3⋅ 3 _{c) 2 5}6⋅ 6 _{d) 20 : 5}4 4

e)

(

3 3

)

3

5 4 : 2⋅ _{f) 36 : 2 9}4

(

4⋅ 4

)

_{g) 6 : 21 : 7} 3

(

3 3

)

_h)

(

_{30 : 5 : 2 3}7 7

) (

5 ⋅ 5

)

Exercise 13

Work out these, giving your answers in index form.

a) 2 4

8 8⋅ _{b) x x}8⋅ 3 _{c) 5 : 5}10 6 _{d) a : a}9 2

e)

( )

4 3

7 f)

( )

5 3

a g)

(

3 5

) (

4

)

a a : a a⋅ ⋅ _h)

(

_{x : x}3 2

) (

⋅ _x4⋅_x3

)

Exercise 14

Compute and answer

Is the square of a sum the same as the sum of the squares of the addends? Is the cube of a sum the same as the sum of the cubes of the addends?

If you generalize the previous results you can say that ___________________

____________________________________________________________

(

)

n n n

a b

⋅

=

a b

⋅

(

a : b

)

n

=

a : b

n n

0

a

=

1

1

a

=

a

2.3.- SQUARES AND SQUARE ROOTS

Remember that a square number is the result of multiplying a whole number by itself.

Example:

9 is a square number because 9 3 3 3= ⋅ = 2

A square number can be represented in the shape of a square.

The perfect squares are the squares of the whole numbers.

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, ….

A square root is a number that when multiplied by itself is equal to a given number. Square roots are written using notation.

2

9 =3 ⇔ 3 =9

You can think of finding square roots as the opposite of finding squares.

You can use trial and improvement to estimate the square root of a number that is not a perfect square.

Example:

Use trial and improvement to find 30 to 1 decimal place.

Estimate Check (square

of estimate) Answer Result

5 2

5 25 Too small

6 ₆ 2 _{36 Too big}

5.5 _5.5 2 _{30.25 Too big}

5.4 2

5.4 29.16 Too small

5.45 2

5.45 29.7025 Too small

So 30 =5.5 _{to 1 decimal place.}

3

3 32 =9

Finding the square root of a number is the

inverse operation of squaring of that number.

a _a 2 _{a = a}2

3 9 9 =3

5 25 25 =5

Exercise 15

The cards show square roots and numbers. Without calculator, match the cards in pairs.

Exercise 16

Without a calculator, write the whole number that is closest in value to

a) 50 b) 80 c) 120 d) 150 e) 8 f) 5

Exercise 17

Two consecutive numbers are multiplied together. The answer is 3192. What are the two numbers?

Exercise 18

Use trial and improvement method to find the square root of the following numbers to 1 decimal place.

a) 20 b) 40 c) 60 d) 95

Exercise 19

The area of a square piece of land is 900 m2. Find the length of its side.

Exercise 20

A tile square wall has two thousand two hundred and nine square tiles. How many rows of tiles are there?

2.4.- CUBES AND CUBE ROOTS

Remember that a cube number is a number that results from multiplying another numbers three times by itself.

Example:

125 is a cube number because 3 125 5 5 5 5= ⋅ ⋅ =

A cube number can be represented in the shape of a cube.

1 444 729 1764 3249

42

38 57 27

3

A cube root is a number that when multiplied by itself three times is equal to a given number. Cube roots are written using 3

notation.

3 3

125 =3 ⇔ 5 =125

You can think of finding cube roots as the opposite of finding cubes.

Exercise 21

In each of these lists of numbers, identify the square and cube numbers.

a) 4, 11, 16, 27, 35 b) 24, 44, 64, 84, 124, 144

c) 156, 196, 216, 256, 286 d) 700, 800, 900, 1000, 1200

Exercise 22

Calculate these using a calculator, giving your answers to 2 dp as appropriate.

a) 3

729 b) 3

100 c) 3_{1 500 d)} 3₁₂₁₆₇

Exercise 23

Calculate the number of small cubes with edges one centimetre in length that contains a cube where each edge is 1 m long.

Finding the cube root of a number is the

inverse operation of cubing of that number.

a _a 3 3_{a = a}3

3 27 3

27 =3

5 25 3

125 =5

10 1000 3