2/(3π⋅7,5⋅10^(-2) )⋅6⋅10^(-10)⋅100⋅(1+0,25)[(1+0,25)^3-0,25^3 ]/(1+2⋅0,25)^2

Let's break down the given expression step by step:

\[
\frac{2}{3\pi \cdot 7.5 \cdot 10^{-2}} \cdot 6 \cdot 10^{-10} \cdot 100 \cdot (1 + 0.25) \left[ (1 + 0.25)^3 - 0.25^3 \right] \div (1 + 2 \cdot 0.25)^2
\]

### Step 1: Simplify the constants and expressions inside the brackets

1. First, calculate the values inside the parentheses:
   - \( 1 + 0.25 = 1.25 \)
   - \( (1.25)^3 = 1.25 \cdot 1.25 \cdot 1.25 = 1.953125 \)
   - \( 0.25^3 = 0.25 \cdot 0.25 \cdot 0.25 = 0.015625 \)

Now substitute the values back into the expression:

\[
\frac{2}{3\pi \cdot 7.5 \cdot 10^{-2}} \cdot 6 \cdot 10^{-10} \cdot 100 \cdot 1.25 \left[ 1.953125 - 0.015625 \right] \div (1 + 2 \cdot 0.25)^2
\]

2. Simplify the subtraction inside the brackets:
   \[
   1.953125 - 0.015625 = 1.9375
   \]

So the expression becomes:

\[
\frac{2}{3\pi \cdot 7.5 \cdot 10^{-2}} \cdot 6 \cdot 10^{-10} \cdot 100 \cdot 1.25 \cdot 1.9375 \div (1 + 2 \cdot 0.25)^2
\]

### Step 2: Simplify the denominator

Now, calculate the term in the denominator:
\[
1 + 2 \cdot 0.25 = 1 + 0.5 = 1.5
\]
\[
(1.5)^2 = 1.5 \cdot 1.5 = 2.25
\]

Now substitute this into the expression:

\[
\frac{2}{3\pi \cdot 7.5 \cdot 10^{-2}} \cdot 6 \cdot 10^{-10} \cdot 100 \cdot 1.25 \cdot 1.9375 \div 2.25
\]

### Step 3: Simplify the remaining expression

We can now focus on simplifying the remaining constants:

- \( 3\pi \approx 9.4248 \)
- \( 7.5 \cdot 10^{-2} = 0.075 \)
- The numerator constants: \( 2 \cdot 6 \cdot 10^{-10} \cdot 100 \cdot 1.25 \cdot 1.9375 \)

Now, let's calculate:

#### Numerator:

\[
2 \cdot 6 = 12
\]
\[
12 \cdot 10^{-10} = 1.2 \cdot 10^{-9}
\]
\[
1.2 \cdot 10^{-9} \cdot 100 = 1.2 \cdot 10^{-7}
\]
\[
1.2 \cdot 10^{-7} \cdot 1.25 = 1.5 \cdot 10^{-7}
\]
\[
1.5 \cdot 10^{-7} \cdot 1.9375 = 2.90625 \cdot 10^{-7}
\]

So the numerator is \( 2.90625 \cdot 10^{-7} \).

#### Denominator:

\[
3\pi \cdot 0.075 = 9.4248 \cdot 0.075 = 0.70686
\]
Now divide by \( 2.25 \):

\[
\frac{0.70686}{2.25} \approx 0.3148
\]

### Step 4: Final calculation

Now, let's finish by dividing the numerator by the denominator:

\[
\frac{2.90625 \cdot 10^{-7}}{0.3148} \approx 9.24 \cdot 10^{-7}
\]

### Final Answer:

\[
\approx 9.24 \cdot 10^{-7}
\]

Would you like any further details or clarifications on specific steps?

---

Here are some related questions to explore:

1. How does the value of \( \pi \) affect the final result?
2. What happens if the power of 10 in the expression changes?
3. How would you approach simplifying an expression with both exponents and trigonometric functions?
4. How does the distributive property apply when simplifying terms with multiple factors?
5. What are some methods for estimating large and small numbers effectively?

**Tip:** When dealing with complex expressions involving exponents, always try to simplify each part step-by-step to avoid errors in calculation.

Learn how to simplify and solve a complex algebraic expression involving exponents and arithmetic operations. This detailed solution demonstrates step-by-step simplification and includes key concepts such as exponentiation, the order of operations, and arithmetic simplifications. Suitable for students in Grades 9-11.

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