Question

If \(x^3 = 8\), then \(x^2(4/(3-x))(2/(4-x)) - (4/x^2) = ?\)

• A. 16
• B. 35
• C. 0
• D. 15
• E. 22

Hint:

Break down complex expressions into smaller, manageable parts. Calculate the value of each parenthesis or fraction separately before combining them. This reduces the chance of arithmetic errors.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem requires first solving a simple cubic equation for a variable and then substituting that value into a more complex algebraic expression to evaluate it.
Step 2: Key Formula or Approach:
1. Solve for x from the equation \(x^3 = 8\). 2. Substitute the value of x into the expression \(x^2(\frac{4}{3-x})(\frac{2}{4-x}) - \frac{4}{x^2}\). 3. Evaluate the expression using the order of operations.
Step 3: Detailed Explanation:
First, find the value of x: \[ x^3 = 8 \] \[ x = \sqrt[3]{8} \] \[ x = 2 \] Now, substitute \(x=2\) into the given expression: \[ 2^2 \left( \frac{4}{3-2} \right) \left( \frac{2}{4-2} \right) - \frac{4}{2^2} \] Evaluate each part of the expression: \[ = 4 \left( \frac{4}{1} \right) \left( \frac{2}{2} \right) - \frac{4}{4} \] \[ = 4(4)(1) - 1 \] Perform the multiplication: \[ = 16 - 1 \] \[ = 15 \] Step 4: Final Answer:
The value of the expression is 15.