Given that 11011 base 2 +11111 base 2 + 10000 is 10m10n0 base 2. Find the values of m and n respectively.
well,
11011 + 11111 + 10000 = 1001010
so ...
To solve this problem, we need to perform addition in base 2. Let's begin by adding the binary numbers 11011 and 11111.
```
11011
+ 11111
________
110010
```
The sum of the binary numbers 11011 and 11111 is 110010.
Next, we need to add this result to the binary number 10000.
```
110010
+ 10000
________
1000010
```
The sum of the binary numbers 110010 and 10000 is 1000010.
We are given that this final result in base 2 is equivalent to 10m10n0. To find the values of m and n, we need to convert this binary number back to decimal.
10m10n0 (base 2) = 1000010 (base 2)
To convert a binary number to decimal, we can use the method of positional notation. Each digit in the binary number represents a power of 2, starting from the rightmost digit.
In our case:
```
1 * 2^6 + 0 * 2^5 + 0 * 2^4 + 0 * 2^3 + 0 * 2^2 + 1 * 2^1 + 0 * 2^0
= 64 + 0 + 0 + 0 + 0 + 2 + 0
= 66
```
Therefore, the decimal equivalent of 1000010 (base 2) is 66.
Now, let's write the decimal number 66 in the form 10m10n0.
66 = 10m10n0
From this equation, we can see that m = 6 and n = 6.
So, the values of m and n respectively are 6 and 6.
To find the values of m and n in the expression 10m10n0 base 2, we can convert the given binary numbers to decimal form and then perform the addition.
11011 base 2 = 1*2^4 + 1*2^3 + 0*2^2 + 1*2^1 + 1*2^0
= 16 + 8 + 0 + 2 + 1
= 27
11111 base 2 = 1*2^4 + 1*2^3 + 1*2^2 + 1*2^1 + 1*2^0
= 16 + 8 + 4 + 2 + 1
= 31
10000 base 2 = 1*2^4 + 0*2^3 + 0*2^2 + 0*2^1 + 0*2^0
= 16 + 0 + 0 + 0 + 0
= 16
So, 11011 base 2 + 11111 base 2 + 10000 = 27 + 31 + 16
= 74
Now let's convert 74 to binary form:
74 = 1*2^6 + 0*2^5 + 1*2^4 + 0*2^3 + 1*2^2 + 0*2^1 + 0*2^0
= 64 + 0 + 16 + 0 + 4 + 0 + 0
= 84 base 2
Thus, the values of m and n in the expression 10m10n0 base 2 are m = 8 and n = 4 respectively.