THE POWER OF 2

The power of 2 is nothing but multiplying 2 by itself.

For example, if the power of 2 is 3, we have to multiply 2 by itself 3 times.

More clearly, 

2³  =  2 x 2 x 2

2³  =  8

The power of 2

Powers of 2 get large very quickly. It’s true that 2¹ is equal to 2 and 2² equals just 4. But 2¹³ equals 8192, which is slightly more than Earth’s diameter in miles !

Let us see, how the value of "powers of 2" gets larger and larger when we increase the powers from 0 to 10

2⁰  =  1

2¹  =  2

2²  =  4

2³  =  8

2⁴  =  16

2⁵  =  32

2⁶  =  64

2⁷  =  128

2⁸  =  256

2⁹  =  512

2¹⁰  =  1024

The Power of 2 - Activity

Freelance Computer Programmer :

David is a freelance computer programmer contracted by a company that makes video games.

In his work, he often used the power of 2ⁿ to find the number of bits (or units of information) that can be arranged. The n stands for the number of bits.

In the video game of hidden treasures that he is programming, the main character wins if he collects 3 fewer hidden treasures than the highest number of bits that can be arranged in the system.

If David is working with a 16-bit system, is the main character a winner if he has collected 65,000 hidden treasures ? Explain your reasoning.

Solution : 

The formula for winning number of hidden treasures is

2ⁿ - 3

Since, David is working with 16-bit system, we have to substitute 16 for 'n' in the above formula.

Then, the winning number of treasures is 

2¹⁶ - 3  =  65536 - 3

2¹⁶ - 3  =  65533

Hence, the main character is not a winner.  Because he has only collected 65,000 hidden treasures which is less than the winning amount 65533.

The Power of 2 - Practice Problems

Problem 1 :

A man has 5 friends. In how many ways, can he invite one or more of his friends to dinner ?

Solution :

We have '2' alternatives for each friend. That is, either he may invite or he may not invite. 

Therefore, no. of all possible ways to invite  5 friends to dinner is

=  2 x 2 x 2 x 2 x 2

=  2⁵

(But it includes the way of not inviting all the 5 friends) 

So, no. of ways to invite one or more of his friends is 

=  2⁵ - 1 

=  32 - 1

=  31

Hence, the no. of ways to invite one more of his friends is 31.

Problem 2 :

A examination paper with 10 questions consists of 6 questions in Algebra and 4 questions in geometry. At least one question is to be attempted from each section. In  how many ways can this be done ?

Solution :

We have '2' alternatives for each question. That is, either we may attempt or we may not attempt. 

Therefore,no. of ways to attempt six questions in Algebra is

=  2 x 2 x 2 x 2 x 2 x 2

=  2⁶

(But it includes the way of not attempting all the questions) 

So, no. of ways to attempt at least one question in Algebra is 

=  2⁶ - 1 

Similarly, no.of ways to attempt atleast one question in Geometry is

= 2⁴ - 1 

Total no.ways for both the sections is

=  (2⁶ - 1)(2⁴ - 1)

=  (64 - 1)(16 - 1)

=  63 x 15

=  945

Hence, the no. of ways of attempting at least one question from each section is 945

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