Problem 91 - Algebra and Trigonometry

Question

Find all values of \(x\) satisfying the given conditions. $$y=(x+4)^{\frac{3}{2}} \text { and } y=8$$

Step-by-Step Solution

Answer

The only value of \(x\) satisfying the given conditions is \(x = 0\).

Step 1: Equating the two functions

First, we need to set the two functions equal to each other to find where they intersect. That can be done by substituting \(y=8\) into the first function. This yields the following equation: \[8 = (x + 4)^{3/2}.\]

Step 2: Solving for \(x\)

To isolate \(x\), we need to get rid of the 3/2 power on the right side of the equation. We can do this by squaring both sides, which yields the equation \[64 = (x + 4)^3.\] Then, taking the cube root of both sides gives us an equation in terms of \(x\): \[(x + 4) = \sqrt[3]{64}\].

Step 3: Simplifying the final equation

Simplifying the equation \((x + 4) = \sqrt[3]{64}\) gives us \(x + 4 = 4\). Subtracting 4 from each side, we then find the value of \(x\), which is \(x = 0\).