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Convert the following expression into binary:
[tex]\[ 4 \times 5^4 + 3 \times 5^3 + 1 \times 5^2 + 0 \times 5^1 + 0 \times 5^0 \][/tex]

Sagot :

GinnyAnswer
Sure! Let's break down the problem step-by-step to find the binary representation of the given expression:

Given the expression:
[tex]\[ 4 \times 5^4 + 3 \times 5^3 + 1 \times 5^2 + 0 \times 5^1 + 0 \times 5^0 \][/tex]

### Step 1: Evaluate the Expression in Decimal Form

First, we need to convert each term in the expression to its decimal value. Here’s how we evaluate each term:
[tex]\[ 4 \times 5^4 = 4 \times 625 = 2500 \][/tex]
[tex]\[ 3 \times 5^3 = 3 \times 125 = 375 \][/tex]
[tex]\[ 1 \times 5^2 = 1 \times 25 = 25 \][/tex]
[tex]\[ 0 \times 5^1 = 0 \times 5 = 0 \][/tex]
[tex]\[ 0 \times 5^0 = 0 \times 1 = 0 \][/tex]

Now, add all these values together:
[tex]\[ 2500 + 375 + 25 + 0 + 0 = 2900 \][/tex]

So, the decimal value of the given expression is [tex]\( 2900 \)[/tex].

### Step 2: Convert Decimal to Binary

Next, we need to convert the decimal number [tex]\( 2900 \)[/tex] to its binary representation. The binary system is a base-2 number system that uses only two symbols: [tex]\( 0 \)[/tex] and [tex]\( 1 \)[/tex].

To convert [tex]\( 2900 \)[/tex] to binary, we repeatedly divide the number by [tex]\( 2 \)[/tex] and record the remainders:

[tex]\[ 2900 \div 2 = 1450 \][/tex] remainder [tex]\( 0 \)[/tex]
[tex]\[ 1450 \div 2 = 725 \][/tex] remainder [tex]\( 0 \)[/tex]
[tex]\[ 725 \div 2 = 362 \][/tex] remainder [tex]\( 1 \)[/tex]
[tex]\[ 362 \div 2 = 181 \][/tex] remainder [tex]\( 0 \)[/tex]
[tex]\[ 181 \div 2 = 90 \][/tex] remainder [tex]\( 1 \)[/tex]
[tex]\[ 90 \div 2 = 45 \][/tex] remainder [tex]\( 0 \)[/tex]
[tex]\[ 45 \div 2 = 22 \][/tex] remainder [tex]\( 1 \)[/tex]
[tex]\[ 22 \div 2 = 11 \][/tex] remainder [tex]\( 0 \)[/tex]
[tex]\[ 11 \div 2 = 5 \][/tex] remainder [tex]\( 1 \)[/tex]
[tex]\[ 5 \div 2 = 2 \][/tex] remainder [tex]\( 1 \)[/tex]
[tex]\[ 2 \div 2 = 1 \][/tex] remainder [tex]\( 0 \)[/tex]
[tex]\[ 1 \div 2 = 0 \][/tex] remainder [tex]\( 1 \)[/tex]

Reading the remainders from bottom to top, we get the binary representation of the decimal number [tex]\( 2900 \)[/tex]:
[tex]\[ 2900_{10} = 101101010100_2 \][/tex]

### Final Answer

The binary representation of the expression
[tex]\[ 4 \times 5^4 + 3 \times 5^3 + 1 \times 5^2 + 0 \times 5^1 + 0 \times 5^0 \][/tex]
is:
[tex]\[ 101101010100_2 \][/tex]
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