Problem 32 - College Algebra

Question

Solve and check each equation with rational exponents. $$ (x+5)^{3 / 2}=8 $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = -1 \)

Step 1: Isolate the term with the rational exponent

Rewrite the equation by isolating the term with the rational exponent on one side: \[ (x+5)^{3 / 2} = 8\]

Step 2: Apply the square root to both sides of the equation

We can take the 2/3 power on both sides to eliminate the 3/2 power on the left hand side. That will result in a linear equation: \[ (x+5) = \sqrt[3]{8^2}=\sqrt[3]{64} \]

Step 3: Solve the Linear Equation

The cube root of 64 is 4, our equation now becomes \( x + 5 = 4 \). Solve this algebraic linear equation to find the solution \(x = 4 - 5 = -1\)

Step 4: Check the solution

To validate the solution, we substitute \(x\) with \(-1\) in the original equation: \[ ((-1) + 5)^{3 / 2} = 8.\] Simplifying this results to \(8 = 8\) indicating that our solution is indeed correct