Sure, let's tackle the problem step-by-step.
### (a) Which law of indices is used to simplify the problem?
To simplify the problem [tex]\(\frac{4.5 \times 10^{25}}{1.5 \times 10^{23}}\)[/tex], we use the law of indices:
[tex]\[
\frac{a^m}{a^n} = a^{m-n}
\][/tex]
This law states that when dividing two powers of the same base, you can subtract the exponents.
### (b) Simplify the given problem
Let's simplify the problem [tex]\(\frac{4.5 \times 10^{25}}{1.5 \times 10^{23}}\)[/tex].
1. Separate the coefficients and the powers of 10:
[tex]\[
\frac{4.5 \times 10^{25}}{1.5 \times 10^{23}} = \frac{4.5}{1.5} \times \frac{10^{25}}{10^{23}}
\][/tex]
2. Simplify the coefficients:
[tex]\[
\frac{4.5}{1.5} = 3
\][/tex]
3. Apply the law of indices to simplify the powers of 10:
[tex]\[
\frac{10^{25}}{10^{23}} = 10^{25-23} = 10^2
\][/tex]
4. Combine the results:
[tex]\[
3 \times 10^2
\][/tex]
Thus, the simplified value is:
[tex]\[
300 \times 10^1 \approx 30000.000000000007
\][/tex]
### (c) Express the result in quinary number system (base 5)
To convert the decimal result into the quinary (base 5) number system, follow these steps for the coefficient part [tex]\(3\)[/tex] (noting that [tex]\(10^2\)[/tex] remains the same as the power does not change with the base):
1. The coefficient part is already [tex]\(3\)[/tex]. In base 5, it is still [tex]\(3\)[/tex], as [tex]\(3\)[/tex] is less than [tex]\(5\)[/tex].
2. Therefore, we express the result as:
[tex]\[
3 \times 10^2 \text{ (in base 5)}
\][/tex]
Hence, the result in quinary (base 5) is:
[tex]\[
3 * 10^2
\][/tex]
So, summarizing:
(a) The law of indices used: [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]
(b) Simplified value: [tex]\(30000.000000000007\)[/tex]
(c) Expressed in quinary number system: [tex]\(3 * 10^2\)[/tex]
