https://r2.erweima.ai/mathbot/image/1725544505872ekez5k6d.jpg

The problem shown in the image involves simplifying the given expression using powers of 10. Let's break it down step by step to explain how to reach the final result:

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

#### Step 1: Rewrite the numbers in scientific notation
- \( 9 \,000 = 9 \times 10^3 \)
- \( 0.000 \,81 = 8.1 \times 10^{-4} \)
- \( 0.000 \,000 \,243 = 2.43 \times 10^{-7} \)

Now, substitute these into the equation:

\[
E = \frac{(9 \times 10^3)^3 (8.1 \times 10^{-4})^2}{(2.43 \times 10^{-7})^2}
\]

#### Step 2: Apply the powers to the terms
- \( (9 \times 10^3)^3 = 9^3 \times (10^3)^3 = 729 \times 10^9 \)
- \( (8.1 \times 10^{-4})^2 = 8.1^2 \times (10^{-4})^2 = 65.61 \times 10^{-8} \)
- \( (2.43 \times 10^{-7})^2 = 2.43^2 \times (10^{-7})^2 = 5.9049 \times 10^{-14} \)

#### Step 3: Combine the terms
Now, substitute these values back into the expression:

\[
E = \frac{729 \times 10^9 \times 65.61 \times 10^{-8}}{5.9049 \times 10^{-14}}
\]

#### Step 4: Simplify the powers of 10
Combine the powers of 10:
- Numerator: \( 729 \times 65.61 \times 10^{9-8} = 47830.89 \times 10^1 = 478308.9 \)
- Denominator: \( 5.9049 \times 10^{-14} \)

So now:

\[
E = \frac{478308.9}{5.9049} \times 10^{1 + 14} = 81 \times 10^{15}
\]

#### Step 5: Final result
\[
E = 81 \times 10^{17}
\]

This is the simplified expression for \( E \), matching the final boxed result in the solution provided in the image.

---

Would you like more details or explanations? Here are some related questions you might find useful:

1. How does scientific notation simplify large or small number calculations?
2. What is the general rule for multiplying and dividing powers of 10?
3. How can exponent rules be applied in real-world calculations?
4. How do powers of numbers interact when they are raised again by another power?
5. What are the common mistakes when simplifying complex scientific notation expressions?

**Tip:** Always ensure you apply the exponent rules correctly when working with scientific notation to avoid mistakes in simplifying.

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

Learn how to simplify expressions involving large and small numbers using scientific notation and exponent rules. The problem demonstrates step-by-step simplification of the expression (9,000)^3 * (0.00081)^2 / (0.000000243)^2, focusing on applying exponent rules and scientific notation.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation