Question

Express 1000 using base 2 and exponents.

Answer 1

Hint:
Binary numbers are also known as base 2 numbers. Binary numbers are written only in ones and zeros. Power is a way of indicating that a number is multiplied by itself a few number of times. For example in ${x^n}$, x is the base and n is the exponent or power. Using this info, express 1000 in base 2 and exponents

Complete step-by-step solution:
We are given to express 1000 using base 2 and exponents.
Using Exponents:
1000 is an even number, so it can also be written as two times 500.
$1000 = 2 \times 500$
500 is an even number, so it can also be written as two times 125.
$
  500 = 2 \times 125 \\
  1000 = 2 \times 2 \times 125 \\
 $
125 is an odd number, and the sum of the digits of 125 is not divisible by 3. Therefore, it is not divisible by 3. Go for the next number. 125 is not divisible by 4. So the next number will be 5.
125 can be written as five times 25.
$
  125 = 5 \times 125 \\
  1000 = 2 \times 2 \times 5 \times 25 \\
 $
25 can be written as five times 5.
$
  25 = 5 \times 5 \\
  1000 = 2 \times 2 \times 5 \times 5 \times 5 \\
 $
1000 is the product of two 2’s and three 5’s. Therefore, the exponent of two is 2 and five is 3.
1000 using exponents is $1000 = {2^2} \times {5^3}$
Using base 2:
To write a number in base 2 means, the number must be a sum of powers of 2.
512 is the 9th power of 2, 256 is the 8th power of 2, 128 is the 7th power of 2, 64 is the 6th power of 2.
Adding these numbers will give $512 + 256 + 128 + 64 = 960$, the rest 40 can be written as the sum of 5th power of 2 and 3rd power of 2.
$
  40 = 32 + 8 = {2^5} + {2^3} \\
  1000 = 960 + 40 \\
  1000 = {2^9} + {2^8} + {2^7} + {2^6} + {2^5} + {2^3} \\
  1000 = {1.2^9} + {1.2^8} + {1.2^7} + {1.2^6} + {1.2^5} + {0.2^4} + {1.2^3} + {0.2^2} + {0.2^1} + {0.2^0} \\
 $
We can observe that the above expression is precisely the binary representation of 1000.
We can write 1000 in base 2 as ${\left( {1111101000} \right)_2}$

Note:
Base 2 numbers are binary numbers and base 10 numbers are normal natural numbers. Binary numbers have only 1’s and 0’s whereas normal numbers can have digits from 0 to 9. So be careful while dealing with the bases.