Let's address each part of the question step-by-step.
### Part (a): Short form of his fee in quinary number
Given the expanded form of the fee:
[tex]\[ 4 \times 5^4 + 3 \times 5^3 + 1 \times 5^2 + 0 \times 5^2 + 0 \times 5^3 \][/tex]
To simplify this to its short form in quinary:
- The coefficients of each [tex]\(5^i\)[/tex] term are the digits in the quinary representation.
- Identify each exponent:
- [tex]\(4\)[/tex] corresponds to [tex]\(5^4\)[/tex],
- [tex]\(3\)[/tex] corresponds to [tex]\(5^3\)[/tex],
- [tex]\(1\)[/tex] corresponds to [tex]\(5^2\)[/tex],
- The two [tex]\(0\)[/tex] entries correspond to the repeated values in [tex]\(5^2\)[/tex] and [tex]\(5^3\)[/tex], but with coefficient [tex]\(0\)[/tex].
So, the short form in quinary is:
[tex]\[ 43100_5 \][/tex]
### Part (b): Convert his monthly fee into a binary number
First, convert the quinary number to a decimal number. The quinary number [tex]\(43100_5\)[/tex] can be converted as follows:
[tex]\[ 4 \times 5^4 + 3 \times 5^3 + 1 \times 5^2 + 0 \times 5^1 + 0 \times 5^0 \][/tex]
This equals:
[tex]\[ 4 \times 625 + 3 \times 125 + 1 \times 25 + 0 \times 5 + 0 \times 1 = 2500 + 375 + 25 + 0 + 0 = 2900 \][/tex]
Now, convert the decimal number [tex]\(2900\)[/tex] into binary:
[tex]\[ 2900_{10} = 101101010100_2 \][/tex]
### Part (c): Find his yearly fee in scientific notation
First, calculate the yearly fee in decimal form. Since his monthly fee is [tex]\(2900\)[/tex] in decimal:
[tex]\[ \text{Yearly fee} = 2900 \times 12 = 34800 \][/tex]
Finally, convert the yearly fee into scientific notation:
[tex]\[ 34800 \text{ in scientific notation} = 3.48 \times 10^4 \][/tex]
Or represented as:
[tex]\[ 3.48e+04 \][/tex]
### Summary
- (a) The short form of his fee in quinary number: [tex]\(43100_5\)[/tex]
- (b) The binary representation of his monthly fee: [tex]\(101101010100_2\)[/tex]
- (c) His yearly fee in scientific notation: [tex]\(3.48e+04\)[/tex]
