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Let's analyze the provided mathematical expression and verify its reduction step-by-step.

The given expression is:

\[
E = \frac{(9 \, 000)^3 (0.000 \, 81)^2}{(0.000 \, 000 \, 243)^2}
\]

To simplify, we can first express these numbers in scientific notation:

1. \(9 \, 000 = 9 \times 10^3\)
2. \(0.000 \, 81 = 81 \times 10^{-5} = 3^4 \times 10^{-5}\)
3. \(0.000 \, 000 \, 243 = 243 \times 10^{-9} = 3^5 \times 10^{-9}\)

Substitute these into the expression for \(E\):

\[
E = \frac{(9 \times 10^3)^3 \cdot (3^4 \times 10^{-5})^2}{(3^5 \times 10^{-9})^2}
\]

Now, let's simplify each term:

1. \((9 \times 10^3)^3 = 9^3 \times (10^3)^3 = (3^2)^3 \times 10^9 = 3^6 \times 10^9\)
2. \((3^4 \times 10^{-5})^2 = 3^8 \times 10^{-10}\)
3. \((3^5 \times 10^{-9})^2 = 3^{10} \times 10^{-18}\)

Now plug these back into the expression:

\[
E = \frac{3^6 \times 10^9 \times 3^8 \times 10^{-10}}{3^{10} \times 10^{-18}}
\]

Combine the powers:

\[
E = \frac{3^{6+8} \times 10^{9-10}}{3^{10} \times 10^{-18}} = \frac{3^{14} \times 10^{-1}}{3^{10} \times 10^{-18}}
\]

Simplify by subtracting exponents:

\[
E = 3^{14-10} \times 10^{-1 - (-18)} = 3^4 \times 10^{17}
\]

Calculate \(3^4\):

\[
3^4 = 81
\]

Thus, 

\[
E = 81 \times 10^{17}
\]

The expression and the steps in the provided solution are correct.

Would you like further details or have any specific questions?

Here are some related questions:
1. How do you convert numbers to scientific notation?
2. What are the rules for simplifying exponents?
3. How can you multiply and divide powers of ten?
4. Why do you need to express numbers in scientific notation for simplification?
5. How do exponent rules apply in mathematical expressions?

**Tip:** Always express large or small numbers in scientific notation to make calculations easier.

Simplify the expression: E = (9,000)^3 * (0.00081)^2 / (0.000000243)^2

Explore the process of simplifying the expression E = (9,000)^3 * (0.00081)^2 / (0.000000243)^2 using scientific notation and exponent rules. This step-by-step solution highlights key algebraic concepts and is ideal for students in Grades 10-12.

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