Understanding and accurately comparing powers of 2 is crucial in various technical and mathematical fields, especially in computer science and digital electronics. In this article, we will explore in detail how to solve and understand a problem involving different powers of 2, including correct and incorrect notations, and provide a solution with an explanation.

Introduction to Powers of 2

Let's begin by defining what we mean by powers of 2. A power of 2 is a number that can be expressed as 2 raised to a certain exponent. For example:

216 2484 24194304 2222

Each of these numbers represents how many times 2 is multiplied by itself. This concept is fundamental in digital systems, where binary numbers and their exponential representations are widely used.

Comparing Powers of 2 with Different Exponents

Let's look at the given problem:

A 216
B 2484
C 24194304
D 2222

Now, we need to compare these powers of 2 and arrange them in ascending order. To do this, we must compare the exponents of 2 in ascending order as well.

The exponents are:

16 (A) 222 (D) 484 (B) 4194304 (C)

Therefore, the correct order is:

A D B C

Thus, the given statement C B D A is incorrect. The correct order should be A D B C, which means:

C B D A → Incorrect
A D B C → Correct

Verification Using Logarithms

Another way to verify this solution is by using logarithms. The logarithm of a power of 2 can be determined using the formula:

log2(2x) x

For the given values:

log2(216) 16 log2(2222) 222 log2(2484) 484 log2(24194304) 4194304

Again, these logarithmic values confirm the ascending order of the exponents:

16 222 484 4194304

Another Problem for Practice: Arranging Binary Numbers

Let's consider another problem:

A 256

We can represent 256 as a power of 2:

256 28

Now, let's compare this with the other numbers:

B 1.8 × 1013 C 1.8 × 1013 D 6.73 × 1066

First, we need to understand the exponential notations:

28 256 1.8 × 1013 1.8 trillion (1,800,000,000,000) 6.73 × 1066 6.73 sextillion (6,730,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000)

Clearly, these values are in the correct order without any need for rearrangement. The correct solution is:

A 256
B C 1.8 × 1013
D 6.73 × 1066

Therefore, the statement 'all are correct' is accurate for these values.

Note: In the given order, A, B, and C have the same value, and D is a significantly larger number.

Conclusion

By understanding and comparing powers of 2 and their exponential representations, we can easily solve such problems. The key is to compare the exponents, and logarithms can help verify the solutions. Always remember that in the digital world, binary numbers and their powers play a crucial role in both theoretical and practical applications.

Frequently Asked Questions (FAQs)

Q: Why are powers of 2 important in digital systems?

A: Powers of 2 are essential in digital systems because they form the basis of binary numbers, which are used in computing, digital electronics, and data representation. Binary systems rely on the simplicity and efficiency of using 2 as the base for calculations.

Q: How can I convert from exponential notation to a power of 2?

A: To convert from a number in exponential notation to a power of 2, you need to understand the decimal number and translate it into a binary format. For example, 256 in decimal is 28 in binary.

Q: What is the significance of logarithms in comparing powers of 2?

A: Logarithms help simplify the comparison of large exponents by reducing them to their base form. This method makes it easier to compare and understand the relative sizes of large powers of 2.