Certainly! Let's address the problem step by step:
### (a) [tex]$a^m$[/tex] लाई [tex]$a^n$[/tex] ले गुणा गर्दा कति हुन्छ ? (What is the product of [tex]$a^m$[/tex] and [tex]$a^n$[/tex]?)
When you multiply two exponential expressions with the same base, you add the exponents. Thus, the product of [tex]\(a^m\)[/tex] and [tex]\(a^n\)[/tex] is given by:
[tex]\[a^m \times a^n = a^{m+n}\][/tex]
### (b) उक्त गुणनफल पत्ता लगाउनुहोस् । (Find the product.)
We need to multiply [tex]\(\left(6.8 \times 10^5\right)\)[/tex] by [tex]\(\left(3.9 \times 10^{-7}\right)\)[/tex].
[tex]\[
(6.8 \times 10^5) \times (3.9 \times 10^{-7}) = 6.8 \times 3.9 \times 10^{5 + (-7)} = 26.52 \times 10^{-2} = 0.2652
\][/tex]
So, the product is:
[tex]\[0.2652\][/tex]
### (c) सो भागफल निकाल्नुहोस्। (Calculate the quotient.)
Next, we divide the product obtained in part (b) by [tex]\(7.8 \times 10^{-4}\)[/tex]:
[tex]\[
\frac{0.2652}{7.8 \times 10^{-4}} = \frac{0.2652}{0.00078} = 340.0
\][/tex]
Therefore, the quotient is:
[tex]\[340.0\][/tex]
### (d) 3400 लाई पञ्च आधार सड्ख्यामा बदल्नुहोस् । (Convert 3400 into quinary number system.)
To convert 3400 into the quinary (base-5) number system, we repeatedly divide the number by 5 and record the remainders:
- [tex]\(3400 \div 5 = 680\)[/tex] remainder [tex]\(0\)[/tex]
- [tex]\(680 \div 5 = 136\)[/tex] remainder [tex]\(0\)[/tex]
- [tex]\(136 \div 5 = 27\)[/tex] remainder [tex]\(1\)[/tex]
- [tex]\(27 \div 5 = 5\)[/tex] remainder [tex]\(2\)[/tex]
- [tex]\(5 \div 5 = 1\)[/tex] remainder [tex]\(0\)[/tex]
- [tex]\(1 \div 5 = 0\)[/tex] remainder [tex]\(1\)[/tex]
Reading the remainders from bottom to top, we get [tex]\(3400_{10} = 102100_5\)[/tex].
So, 3400 in the quinary number system is:
[tex]\[102100_5\][/tex]
### (e) 34 लाई द्विआधार सड्ख्यामा बदल्नुहोस् । (Convert 34 into binary number system.)
To convert 34 into the binary (base-2) number system, we repeatedly divide the number by 2 and record the remainders:
- [tex]\(34 \div 2 = 17\)[/tex] remainder [tex]\(0\)[/tex]
- [tex]\(17 \div 2 = 8\)[/tex] remainder [tex]\(1\)[/tex]
- [tex]\(8 \div 2 = 4\)[/tex] remainder [tex]\(0\)[/tex]
- [tex]\(4 \div 2 = 2\)[/tex] remainder [tex]\(0\)[/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 [tex]\(34_{10} = 100010_2\)[/tex].
So, 34 in the binary number system is:
[tex]\[100010_2\][/tex]
