### Part (a):
#### Simplify:
[tex]\[
\frac{1.5 \times 10^4 \times 4.5 \times 10^3}{2.5 \times 10^2}
\][/tex]
Step 1: Multiply the numbers in the numerator (both coefficients and powers of 10).
Coefficients:
[tex]\[
1.5 \times 4.5 = 6.75
\][/tex]
Powers of 10:
[tex]\[
10^4 \times 10^3 = 10^{4+3} = 10^7
\][/tex]
So the numerator becomes:
[tex]\[
6.75 \times 10^7
\][/tex]
Step 2: Simplify the denominator.
The denominator is:
[tex]\[
2.5 \times 10^2
\][/tex]
Step 3: Divide the coefficients and subtract the exponents of 10 in the denominator from those in the numerator.
Coefficients:
[tex]\[
\frac{6.75}{2.5} = 2.7
\][/tex]
Powers of 10:
[tex]\[
10^7 \div 10^2 = 10^{7-2} = 10^5
\][/tex]
Step 4: Combine the results.
So, the simplified result is:
[tex]\[
2.7 \times 10^5
\][/tex]
Converting to standard notation:
[tex]\[
2.7 \times 10^5 = 270,000
\][/tex]
### Part (b):
#### Simplify:
[tex]\[
\frac{6.4 \times 10^{-5} \times 3.6 \times 10^9}{1.6 \times 10^{10} \times 1.8 \times 10^{-3}}
\][/tex]
Step 1: Multiply the numbers in the numerator (both coefficients and powers of 10).
Coefficients:
[tex]\[
6.4 \times 3.6 = 23.04
\][/tex]
Powers of 10:
[tex]\[
10^{-5} \times 10^9 = 10^{-5+9} = 10^4
\][/tex]
So the numerator becomes:
[tex]\[
23.04 \times 10^4
\][/tex]
Step 2: Multiply the numbers in the denominator (both coefficients and powers of 10).
Coefficients:
[tex]\[
1.6 \times 1.8 = 2.88
\][/tex]
Powers of 10:
[tex]\[
10^{10} \times 10^{-3} =10^{10-3} = 10^7
\][/tex]
So the denominator becomes:
[tex]\[
2.88 \times 10^7
\][/tex]
Step 3: Divide the coefficients and subtract the exponents of 10 in the denominator from those in the numerator.
Coefficients:
[tex]\[
\frac{23.04}{2.88} = 8
\][/tex]
Powers of 10:
[tex]\[
10^4 \div 10^7 = 10^{4-7} = 10^{-3}
\][/tex]
Step 4: Combine the results.
So, the simplified result is:
[tex]\[
8 \times 10^{-3}
\][/tex]
Converting to standard notation:
[tex]\[
8 \times 10^{-3} = 0.008
\][/tex]
### Final Answer:
- Part (a): [tex]\( 270,000 \)[/tex]
- Part (b): [tex]\( 0.008 \)[/tex]
