Problem 92 - Algebra and Trigonometry

Question

Find all values of \(x\) satisfying the given conditions. $$y=(x-5)^{\frac{3}{2}} \text { and } y=125$$

Step-by-Step Solution

Answer

The value of \( x \) that satisfies both equations is 10.

Step 1: Equate the two equations

Since \( y \) is common in both equations, equate them to each other: \[ (x-5)^{\frac{3}{2}} = 125 \]

Step 2: Solve for \( x \)

First, take cube root on both sides of the equation which gets rid of the fractional exponent:\[ (x-5) = \sqrt[3]{125} \]We know that \(\sqrt[3]{125}\) is 5, so simplify it to get:\[ x-5 = 5 \]

Step 3: Find the value of \( x \)

Finally, solve for \( x \) by adding 5 to both sides: \[ x = 5+5 = 10 \]