Explanation

We know the identity:

a³ - b³ = (a - b)(a² + ab + b²)

Here, 8x³ - 27y³ = (2x)³ - (3y)³

= (2x - 3y)((2x)² + (2x)(3y) + (3y)²)

Given 2x - 3y = 8 and xy = 2:

(2x)² + (2x)(3y) + (3y)² = 4x² + 6xy + 9y²

= 4x² + 6(2) + 9y²

= 4x² + 12 + 9y²

We need to express 4x² + 9y² in terms of 2x - 3y and xy:

(2x - 3y)² = 4x² - 12xy + 9y²

64 = 4x² - 12(2) + 9y²

64 = 4x² - 24 + 9y²

4x² + 9y² = 88

Now, 4x² + 6xy + 9y² = 88 + 12 = 100

So, 8x³ - 27y³ = (2x - 3y)((2x)² + (2x)(3y) + (3y)²)

= 8 × 100

= 800

Explanation

The fourth root of 256 is:

⁴√256 = ⁴√(4^4) = 4

Explanation

To find the weight of rice per packet:

1350 kg / 30 packets = 45 kg/packet

For 38 packets:

Weight = 38 packets × 45 kg/packet

= 1710 kg

Explanation

Given the volume of the cube-shaped pool is 10648 m³. To find the side length (s):

s³ = 10648

s = ∛10648

s = 22

The area of the base floor = s² = 22² = 484 m²

Cost = Area × Rate

= 484 × 85

= 41140

Explanation

The cube of 12 is:

12 × 12 × 12 = 1728

Explanation

The formula for the volume of a hemisphere is:

(2/3) × π × r^3

Given diameter = 16.8 cm, so radius (r) = 16.8 / 2 = 8.4 cm

Volume = (2/3) × π × (8.4)^3

= (2/3) × 3.14159 × 592.704

≈ 1241.85 cm^3

Explanation

First, let's find the work rate of 10 men:

10 men construct 75 km in 5 days, so:

Work rate = 75 km / 5 days = 15 km/day for 10 men

Now, let's find the work rate for 15 men:

Work rate for 15 men = (15/10) × 15 km/day = 22.5 km/day

Now, let's find the time needed for 15 men to construct 45 km:

Time = Distance / Work rate

= 45 km / 22.5 km/day

= 2 days

Explanation

Distance = Speed × Time

= 102 km/hr × 12 hours

= 1224 km

Now, let's find the time it takes to cover the same distance at 96 km/hr:

Time = Distance / Speed

= 1224 km / 96 km/hr

= 12.75 hours

To convert hours to minutes:

12.75 hours × 60 minutes/hour = 765 minutes

Explanation

The sum of the interior angles of a polygon is given by:

(n - 2) × 180

where n is the number of sides.

For a 5-sided polygon (pentagon):

(5 - 2) × 180 = 3 × 180 = 540

Since it's a regular polygon, all interior angles are equal. To find each angle:

540 ÷ 5 = 108°

Explanation

To find the median, we need to arrange the data in order:

3, 6, 8, 9, 10, 12, 14, 14, 16

Since there are 9 numbers (an odd number), the median is the middle value, which is the 5th number:

The median is 10.