To solve this equation, let's analyze the expression \((1 + x)^7 = 7.054545455\).

### Step 1: Understand the Equation
The equation \((1 + x)^7 = 7.054545455\) suggests that we need to find the value of \(x\) such that when \(1 + x\) is raised to the power of 7, the result equals approximately 7.054545455.

### Step 2: Solve for \(x\)

To find \(x\), we will take the 7th root of both sides:

\[
1 + x = (7.054545455)^{1/7}
\]

Now, let's compute this root to determine the value of \(1 + x\), and subsequently, solve for \(x\). 

### Step 3: Compute the Value

Let's calculate \((7.054545455)^{1/7}\).The 7th root of 7.054545455 is approximately \(1.321934\).

### Step 4: Solve for \(x\)

Now, since \(1 + x = 1.321934\), we can solve for \(x\):

\[
x = 1.321934 - 1 = 0.321934
\]

### Conclusion

The value of \(x\) that satisfies the equation \((1 + x)^7 = 7.054545455\) is approximately \(x \approx 0.321934\).

Would you like more details on any step, or do you have any questions?

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Here are some related questions:
1. How do you calculate the nth root of a number?
2. What are some practical applications of binomial expansion in mathematics?
3. How does raising a number to a fractional power work?
4. Can you explain the properties of exponents and roots?
5. How can logarithms be used to solve exponential equations?

**Tip:** When dealing with powers and roots, remember that raising a number to a fractional exponent is equivalent to taking the root of the number!

Learn how to solve the exponential equation (1+x)^7 = 7.054545455 with a step-by-step solution. This problem involves exponentiation and root calculation, providing a detailed method to find the value of x. Suitable for students in Grades 10-12.

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