Explanation

Let's convert the binary numbers to decimal:

(1000)₂ = 1 × 2³ = 8

(0010)₂ = 0 × 2³ + 1 × 2¹ = 2

Now, multiply:

8 × 2 = 16

Explanation

Let's evaluate the expression:

67.5 × 10⁻⁹ = 6.75 × 10 × 10⁻⁹

= 6.75 × 10⁻⁸

= 0.0000000675

Explanation

Let's simplify the expression:

(((3)²)²)² = (3²)⁴ = 3⁸

Now, divide by 9 (which is 3²):

3⁸ / 3² = 3⁶

Explanation

Let's analyze the options:

1. 1.66666666666666…: This is a repeating decimal, which means it's a rational number (can be expressed as a fraction, 5/3).
2. 0.02002000200002…: This appears to be a non-repeating, non-terminating decimal, which is a characteristic of irrational numbers.
3. 0.142857142857: This is a repeating decimal, which means it's a rational number (can be expressed as a fraction, 1/7).

Based on the analysis, the answer is 0.02002000200002….

Explanation

Let's calculate the total amount of water:

Two full bottles: 2 × 600 ml = 1200 ml

The third bottle (2/3 full): 2/3 × 600 ml = 400 ml

Total water = 1200 ml + 400 ml = 1600 ml

Explanation

A rational number can be written as a fraction (p/q) with both p and q as integers.

All given options are irrational:

3.129765917723... is non-repeating, non-terminating

π and √2 are well-known irrational numbers.

Explanation

Let's evaluate each option:

1. sin² 60⁰ + cos² 60⁰ = 1

sin 60⁰ = √3/2, cos 60⁰ = 1/2

(√3/2)² + (1/2)² = 3/4 + 1/4 = 1

2. sin² 45⁰ + cos² 45⁰ = 1

sin 45⁰ = cos 45⁰ = 1/√2

(1/√2)² + (1/√2)² = 1/2 + 1/2 = 1

3. sin² 45⁰ − cos² 45⁰ = 0

(1/√2)² - (1/√2)² = 1/2 - 1/2 = 0

Explanation

Let's simplify the expression:

∜256a⁴b⁴ = ∜(4⁴ × a⁴ × b⁴)

= ∜(4a × 4a × 4a × 4a × b × b × b × b)

= 4ab

Explanation

The total length of the bridge is:

10 sections × 2.5 meters/section = 25 meters

To find the distance from the edge to the center:

Center = Total length / 2

= 25 meters / 2

= 12.5 meters

Explanation

The correct logarithmic form of x = a^y is:

logₐx = y

This is because logarithms and exponentials are inverse operations.