On a chapter test, Stephanie uses the rules of equation solving and inverse operations to solve the cubic equation below. The directions on the test are not very specific; however, they do say to give an exact solution. Stephanie is not sure if she should submit a decimal approximation or a simplified radical as the answer. 4x^3 = 756

Question

What recommendation can you make to Stephanie regarding her answer to the cubic equation above? In your recommendation, explain what the term "exact" indicates with regards to the solution.
Solve the cubic equation from Stephanie’s test. Submit two solutions for the equation- a) your answer as a decimal approximation, rounded to the nearest hundredth, and b) as a simplified radical. Use substitution to check each version of your solution in the original equation.
After solving the test question and checking both the decimal solution and the simplified radical form, would you change your recommendation (in Part 1) to Stephanie regarding the format of her answer? Use complete sentences to explain your reasoning.

WRITER

Answer 1

Recommendation to Stephanie:

I would recommend that Stephanie submit her solution in simplified radical form rather than a decimal approximation. The term "exact" typically refers to solutions that are expressed in a form that is fully simplified and does not rely on numerical approximations. In this case, a simplified radical form provides a precise representation of the solution without rounding errors, making it the more mathematically rigorous choice.

Solving the cubic equation:

The equation Stephanie is solving is: \[ 4x^3 = 756 \]

To solve for \( x \), first divide both sides by 4: \[ x^3 = \frac{756}{4} \] \[ x^3 = 189 \]

Now, take the cube root of both sides: \[ x = \sqrt[3]{189} \]

To simplify \( \sqrt[3]{189} \), we can factor 189: \[ 189 = 3^3 \cdot 7 \] This means we can write: \[ \sqrt[3]{189} = \sqrt[3]{3^3 \cdot 7} = 3\sqrt[3]{7} \]

So in simplified radical form, the solution is: \[ x = 3\sqrt[3]{7} \]

Calculating the decimal approximation:

Now, let's find the decimal approximation of \( 3\sqrt[3]{7} \): \[ \sqrt[3]{7} \approx 1.913 (rounded to three decimal places) \]

Thus, the decimal solution is: \[ x \approx 3 \times 1.913 \approx 5.739 \]

Rounding this to the nearest hundredth gives: \[ x \approx 5.74 \]

Verification of both solutions:

Checking the simplified radical solution: Substituting \( x = 3\sqrt[3]{7} \) back into the original equation:

1. Calculate \( 4x^3 \): \[ 4(3\sqrt[3]{7})^3 = 4(27 \cdot 7) = 4 \cdot 189 = 756 \] This confirms that \( 4x^3 = 756 \).

Checking the decimal approximation: Substituting \( x \approx 5.74 \):

1. Calculate \( 4(5.74^3) \): \[ 5.74^3 \approx 189.052 (approx) \] \[ 4 \cdot 189.052 \approx 756.208 \] This does not match 756 exactly due to rounding.

Conclusion on recommendation: Based on the verification, I would maintain my recommendation to Stephanie to submit her answer as a simplified radical \( x = 3\sqrt[3]{7} \) rather than the decimal approximation \( x \approx 5.74 \). Although the decimal is useful for understanding the magnitude of the solution, the exact simplified radical form accurately represents the solution with full precision, adhering to the expectation for an exact answer as specified in the test instructions.