To determine the value of the expression [tex]\(\left(\frac{\left(10^4\right)\left(5^2\right)}{\left(10^3\right)\left(5^3\right)}\right)^3\)[/tex], we need to follow a step-by-step approach to solve it systematically.
1. Identify and calculate the individual components:
- [tex]\(10^4 = 10000\)[/tex]
- [tex]\(5^2 = 25\)[/tex]
- [tex]\(10^3 = 1000\)[/tex]
- [tex]\(5^3 = 125\)[/tex]
2. Form the numerator and the denominator of the fraction inside the parentheses:
- Numerator: [tex]\(10^4 \times 5^2 = 10000 \times 25 = 250000\)[/tex]
- Denominator: [tex]\(10^3 \times 5^3 = 1000 \times 125 = 125000\)[/tex]
3. Calculate the intermediate result of the fraction inside the parentheses:
- [tex]\(\frac{250000}{125000} = 2\)[/tex]
4. Raise the intermediate result to the power of 3:
- [tex]\(2^3 = 2 \times 2 \times 2 = 8\)[/tex]
Therefore, the value of the expression [tex]\(\left(\frac{\left(10^4\right)\left(5^2\right)}{\left(10^3\right)\left(5^3\right)}\right)^3\)[/tex] is:
[tex]\[
\boxed{8}
\][/tex]
