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.

Question

4x^3=756
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.

Answer 1

Recommendation to Stephanie

I recommend that Stephanie submits her answer in the simplified radical form rather than a decimal approximation. The term "exact" in mathematical solutions generally refers to expressing answers in their simplest form, without rounding or approximating values. This often means using radicals or exact integers rather than decimal representations, which involve rounding errors and can lose precision.

Solving the Cubic Equation \(4x^3 = 756\)

First, we can isolate \(x^3\):

\[ x^3 = \frac{756}{4} \]

\[ x^3 = 189 \]

Next, we find \(x\) by taking the cube root of both sides:

\[ x = \sqrt[3]{189} \]

To find simplified radical form, we can factor \(189\):

\[ 189 = 3^3 \times 7 \]

Thus, we have:

\[ x = \sqrt[3]{3^3 \times 7} = \sqrt[3]{3^3} \times \sqrt[3]{7} = 3\sqrt[3]{7} \]

Final Answers:

• a) Decimal approximation: We can compute \( \sqrt[3]{189} \) using a calculator:

\[ x \approx 5.78 \text{ (rounded to the nearest hundredth)} \]

• b) Simplified radical form:

\[ x = 3\sqrt[3]{7} \]

Checking the Solutions

1. Checking the Decimal Solution:

Substituting \(x \approx 5.78\) back into the original equation:

\[ 4(5.78)^3 \approx 4(193.99) \approx 775.96 \text{ (not exactly 756 due to rounding)} \]

2. Checking the Simplified Radical:

For the simplified radical solution:

\[ x = 3\sqrt[3]{7} \] Substituting back:

\[ 4(3\sqrt[3]{7})^3 = 4 \times 27 \times 7 = 4 \times 189 = 756 \]

Since this checks out, we confirm the solution is correct.

Conclusion

Given the results from the checks, I would maintain my recommendation to submit the answer in simplified radical form. The radical form provides an exact value that correctly solves the original equation, while the decimal approximation, while useful, does not represent the exact value due to the inherent round-off in calculations. Therefore, for an exact solution, \(3\sqrt[3]{7}\) is preferred.

Answer 2

can you do it without all the symbols and slashes

Answer 3

Sure! Let's explain the solution without using the mathematical symbols and slashes.

Recommendation to Stephanie

I recommend that Stephanie submits her answer in the form of a simplified radical instead of a decimal approximation. The term "exact" in mathematics usually means providing answers in their simplest form, using whole numbers or radicals rather than decimal numbers. Decimal approximations can involve rounding, which may result in a loss of precision.

Solving the Cubic Equation

To solve the equation 4 times x cubed equals 756, we first isolate x cubed. We do this by dividing both sides of the equation by 4. This gives us x cubed equals 189.

Next, we find the value of x by taking the cube root of 189. To express this in a simplified radical form, we can break down 189 into its factors. The number 189 can be factored into three cubed times seven.

So, taking the cube root, we find that x equals three times the cube root of seven.

Final Answers:

• A) The decimal approximation: After calculating the cube root of 189, we find that x is approximately 5.78 when rounded to the nearest hundredth.
• B) The simplified radical form: This is three times the cube root of seven.

Checking the Solutions

1. Checking the Decimal Solution:

When we substitute 5.78 back into the original equation, while doing the calculation, we find that 4 times 5.78 cubed gives us about 775.96. This is not exactly 756 due to rounding in the decimal approximation.

2. Checking the Simplified Radical:

Now, let's check the simplified radical solution. When substituting three times the cube root of seven back into the original equation, we find that 4 times three cubed times seven calculates to 756 exactly. This means the simplified radical form is correct.

Conclusion

Based on the checks, I encourage Stephanie to stick with the recommendation of submitting the answer in simplified radical form. This version is exact and accurately satisfies the original equation. In contrast, while the decimal approximation is useful, it does not provide the precise value of the solution due to rounding. Therefore, the correct answer is three times the cube root of seven.

Answer 4

can you make it less advanced and less words

Answer 5

Of course! Here’s a simpler and more concise explanation:

Recommendation to Stephanie

I suggest that Stephanie submits her answer as a simplified radical rather than a decimal. "Exact" means to give a precise answer without rounding, like using radicals.

Solving the Equation

To solve the equation "4 times x cubed equals 756":

1. Divide both sides by 4:

   • Now we have "x cubed equals 189."

2. To find x, take the cube root of 189:

   • This can be written as "x equals the cube root of 189."

3. We can simplify the cube root:

   • 189 can be factored into 27 and 7, where 27 is three cubed.
   • So, "x equals 3 times the cube root of 7."

Final Answers:

• A) Decimal: The cube root of 189 is about 5.78 (rounded to two decimal places).
• B) Simplified radical: The answer is 3 times the cube root of 7.

Checking the Solutions

1. Decimal Check:

   • If we plug 5.78 back into the original equation, we get about 775.96, which is not exactly 756.

2. Radical Check:

   • Plugging "3 times the cube root of 7" into the equation gives us exactly 756, which is correct.

Conclusion

I believe Stephanie should stick with the simplified radical answer. It is exact and solves the equation correctly, while the decimal is not precise due to rounding. So, the answer is 3 times the cube root of 7.