We are given the formula:

$E = 0.2 \cdot x^{\frac{3}{2}}$

where:

• $E$ is the number of Earth days in a planet's year.
• $x$ is the average distance from the center star in millions of kilometers.

Step 1: Substitute the given value of $E$

We are given that $E = 224.3$, so we set up the equation:

$224.3 = 0.2 \cdot x^{\frac{3}{2}}$

Step 2: Solve for $x$

a) Isolate $x^{\frac{3}{2}}$

$x^{\frac{3}{2}} = \frac{224.3}{0.2}$

$x^{\frac{3}{2}} = 1121.5$

b) Solve for $x$

To get $x$, raise both sides to the power of $\frac{2}{3}$:

$x = (1121.5)^{\frac{2}{3}}$

Now, let's compute the value. Compute x = (1121.5)^(2/3) x = 1121.5**(2/3) x### Step 3: Round to the nearest whole number

$x \approx 108$

Thus, the average distance of Planet B from the center star is 108 million kilometers.

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Would you like a step-by-step breakdown of exponent manipulation? Here are some related questions to explore:

1. How does the exponent $\frac{3}{2}$ affect the equation?
2. What are the steps to solving equations with fractional exponents?
3. How is this formula related to Kepler’s Third Law?
4. What would happen if the coefficient 0.2 changed?
5. Can we use logarithms to solve this equation?

Tip: When solving equations with exponents, always isolate the power term first before applying inverse operations.