Find the smallest number by which each of the following numbers must be multiplied to obtain a perfect cube.

(i) 243

(ii) 256

(iii) 72

(iv) 675

(v) 100

ANSWER:

(i) 243 = 3 × 3 × 3 × 3 × 3

Here, two 3s are left which are not in a triplet. To make 243 a cube, one more 3 is required.

In that case, 243 × 3 = 3 × 3 × 3 × 3 × 3 × 3 = 729 is a perfect cube.

Hence, the smallest natural number by which 243 should be multiplied to make it a perfect cube is 3.

(ii) 256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

Here, two 2s are left which are not in a triplet. To make 256 a cube, one more 2 is required.

Then, we obtain

256 × 2 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 512 is a perfect cube.

Hence, the smallest natural number by which 256 should be multiplied to make it a perfect cube is 2.

(iii) 72 = 2 × 2 × 2 × 3 × 3

Here, two 3s are left which are not in a triplet. To make 72 a cube, one more 3 is required.

Then, we obtain

72 × 3 = 2 × 2 × 2 × 3 × 3 × 3 = 216 is a perfect cube.

Hence, the smallest natural number by which 72 should be multiplied to make it a perfect cube is 3.

(iv) 675 = 3 × 3 × 3 × 5 × 5

Here, two 5s are left which are not in a triplet. To make 675 a cube, one more 5 is required.

Then, we obtain

675 × 5 = 3 × 3 × 3 × 5 × 5 × 5 = 3375 is a perfect cube.

Hence, the smallest natural number by which 675 should be multiplied to make it a perfect cube is 5.

(v) 100 = 2 × 2 × 5 × 5

Here, two 2s and two 5s are left which are not in a triplet. To make 100 a cube, we require one more 2 and one more 5.

Then, we obtain

100 × 2 × 5 = 2 × 2 × 2 × 5 × 5 × 5 = 1000 is a perfect cube

Hence, the smallest natural number by which 100 should be multiplied to make it a perfect cube is 2 × 5 = 10.

Find out the smallest number multiplied by the following numbers to get the perfect cube? 

(i) 5400

(ii) 10584

(i) 5400 

Resolving 5400 into Prime factors as follows

Here, factor 5 is not ¡n group of three. 

∴ To make 5400 a perfect cube, we have to multiply it by smallest number 5.

(ii) 10584 

Resolving 10584 into Prime factors as follows

Here , factor 7 is not in group of three 

∴ To make 10584 as a perfect cube, we have to multiply it by smallest number 7.

• The moon changes its shape everyday over a period one month and this repeats over and over again. These different shapes of the moon are called 'Phases of the Moon'. In reality, the moon does not change its shape at all. Moon is always round in shape. As we know the moon is always visible because it reflects the light from the sun.
• It goes from a perfect round circle and gradually reduces in size and further becomes a  no moon (as shown here). This takes about 15 days.  This is called waxing.
• Then the moon again begins to grow in size until it becomes a full moon. This takes another 15 days. This is called waning.
• The total span between 2 full moons is approximately 30 days nearly a month.
• At any point of time, the sun is able to illuminate only half part of the moon. The other half of the moon does not receive any portion of sunlight because of its spherical shape.
• Depending on the these positions of sun, moon, and the earth, different proportions of the illuminated moon faces the earth, causing the shape of the moon appearing to change as seen from earth.
• On a full moon day, when the moon appears as a bright complete circle, the earth is directly between the moon and the sun. The illuminated surface of the moon fully faces the earth, making the moon appear round.
• On a new moon phase, the moon is between the Sun and Earth. This means its light is not reflected to Earth. Instead, the light from the sun bounces back from the earth and causes a slight illumination on the moon.
• All shapes or the phases of the moon indicate  its position from the Sun and earth and how light behaves in relation to each.

  Mars (Mangal): It is the first outside the orbit of the earth. It is also known as red planet. It has two satellites.

 Jupiter (Brihaspati): It is the largest planet of the solar system. It has a large number of satellites (moons).

 Saturn (Shani): It appears yellowish in colour. It has beautiful rings. Saturn has a large number of satellites. It is least dense among all the planets.

 Uranus (Indra): Like Venus, Uranus also rotates from east to west. It has highly tilted rotational axis. So in its orbital motion Uranus appears to roll on its side.

 Neptune (Varun): It is the outermost planet of the solar system. It has two moons.

 Mercury (Budh): It is the smallest planet and nearest to the sun. Mercury has no satellite of its own.

        Venus (Shukra): It is nearest to the earth and is the brightest planet in the night sky. Venus has no moon or satellite of its own. It rotates from east to west while the earth rotates from west to east.

The characteristics of eight planets are:
1)Venus : Its surface is a brownish yellow colour because of its atmosphere which is composed of largely of sulphuric acid and CO2.
2)Mars: It takes 687 days to revolve around the sun.This planet is very hottest planet then other planets.
3)Uranus: It rotates the sun in 84 years, its size is 51,118 kilometers in diameter.
4)Neptune: It's size is 49,500 kilometres in diameter. It orbit the sun every 165 years.
5)Mercury:It is the closest planet to the sun,so its orbits the sun very quickly just in 88 days.It surface is grey to orange in colour and it is covered with craters.
6)Earth: Earth rotates the sun every 365 days(years) and rotates in its every axis in 24 hours(day). It size is a little more than 12000 km. In diameter. This planet is the different from the other planets as it maintains life on it.
7)Jupiter:It is the largest of the solar system as a diameter of 142,980.
8)Saturn:It is a large planet as 120,536.It's orbit the sun in 12 years.

Orbital period

The orbital period is the time a given astronomical object takes to complete one orbit around another object, and applies in astronomy usually to planets or asteroids orbiting the Sun, moons orbiting planets

rotation period 

The rotation period of a celestial object (e.g., star, gas giant, planet, moon, asteroid) is the time that the object takes to complete a single revolution around its axis of rotation relative to the background stars. 

Since the planets are much nearer to us than the stars, they appear to be big and do not twinkle at night. The stars appear to twinkle at night but the planets do not twinkle at all. The planets move around the Sun from west to east, so the relative positions of the planets in the night sky keep changing day by day.

Router  acts as a traffic controller for data to and from the internet

Step-by-step explanation:

let the shorter side be x cm

other side = x+1 cm

height = 4cm

Area of trapezium = (1/2)*sum of parallel side* height

                              = (1/2)*(x cm + x+1 cm)*4 cm

                              = 2x+1 cm * 2 cm

                              = 4x+2 cm square

                                       ATQ

       4x+2 = 10

 => 4x = 10 -2

 => x = 8/4

 => x = 2

side 1 = x cm

         = 2 cm

side 2 = x+1 cm

          = 2+1 cm

          = 3 cm

Area of a trapezium = 1/2×height (sum of parallel sides
given:the other parallel side is double the other. height is5cm.Area is 45 cm square.
let the parallel sides be x and 2x
1/2×5 (x+2x)=45
5 (x+2x)=45×2
5x+10x=90
15x=90
x=90/15
x=6
2x=12
so the parallel sides are 6cm and 12cm.

area of trapezium=1/2×(sum of parallel sides)×height

area of trapezium=1/2×(4.1+10.9)×5
area of trapezium=1/2×75
area of trapezium=37.5

B =x
L= x+7

Original Area =x×(x×7)=x^2+7x
New length =x+7-4=x+3
New breadth =x+3
New area =(x+3)(x+3)=x^2+6x+9
A2Q
=x^2+7x =x^2+6x+9
=x^2- x^2+7x=6x+9
=7x-6x=9
=x=9cm=Breadth
9+7=16cm=Length

let the lengths of two sides of the parallelogram be 4x cm and 3x cm respectively

⇒ its perimeter = 2(4x + 3x) cm = 8x + 6x = 14x cm. 

∴ 14x = 210 

⇔ x = 210/14 = 15

∴ one side = (4 × 15) cm = 60 cm and other side = (3 ×15) cm = 45 cm. 

1 2/3+2 3/4+1 1/2

= 5/3 + 11/4 + 3/2

= [5(4) + 11(3) + 3(6)]/ 12

= (12 + 33 + 18)/ 12

= 63/12

Wildlife sanctuary is the place where wild animals are protected in their natural habitat. Here poaching or capturing the animals is prohibited.

 Biosphere reserves are the area used to conserve biodiversity like plants, animals and microorganisms. It contains many wildlife sanctuaries and national parks.