Problem 19 - A Graphical Approach to College Algebra

Question

Solve each equation by hand. Do not use a calculator. $$\sqrt[4]{x-2}+4=6$$

Step-by-Step Solution

Answer

The solution is \( x = 18 \).

Step 1: Isolate the Radical Expression

Start by isolating the fourth root term in the equation. The equation is \( \sqrt[4]{x-2} + 4 = 6 \). Subtract 4 from both sides to get \( \sqrt[4]{x-2} = 2 \).

Step 2: Remove the Fourth Root

Eliminate the fourth root by raising both sides of the equation to the power of four. This results in \( (\sqrt[4]{x-2})^4 = 2^4 \), simplifying to \( x - 2 = 16 \).

Step 3: Solve for x

Add 2 to both sides of the equation to solve for \(x\). This gives \( x = 16 + 2 \), which simplifies to \( x = 18 \).

Step 4: Verify the Solution

Substitute the solution back into the original equation to verify. The original equation is \( \sqrt[4]{18 - 2} + 4 = 6 \). Simplify \( \sqrt[4]{16} + 4 = 6 \). Since \( \sqrt[4]{16} = 2 \) and \( 2 + 4 = 6 \), the solution is verified.

Key Concepts

Fourth Roots

Fourth roots are a type of radical expression, much like square roots, but instead of finding a number that multiplies by itself once to reach the original number, you're looking for a number which, when multiplied by itself four times, reaches the original number. For example, the fourth root of 16 is 2 because

• 2 multiplied by itself is 4
• 4 multiplied by 2 is 8
• 8 multiplied by 2 is 16

This is symbolized by the expression \( \sqrt[4]{x} \), where you're solving to find the value of \( x \) under the root. When working with radical equations that involve fourth roots, the goal is often to isolate the fourth root, and then eliminate it by raising both sides of the equation to the power of 4, effectively 'undoing' the radical. So, understanding how and why the fourth root acts as the inverse of exponentiation by four is key to grasping these types of problems.

Equation Verification

Equation verification is an essential part of solving any mathematical problem, especially equations. It means checking to ensure that your solution is correct by plugging it back into the original equation.
Why is this important? Because it confirms that the solution works and fits with the problem as intended. Getting the right solution is not enough if it doesn't satisfy the original equation.
Here's a simple process to verify an equation:

• Plug the calculated solution back into the original equation.
• Check whether both sides of the equation result in a true statement after substituting the solution.
• If both sides of the equation equal, the solution is verified.

In our case, after solving for \( x = 18 \), we substitute back into the equation to confirm that it holds true: \( \sqrt[4]{18-2} + 4 = 6 \) simplifies to \( 2 + 4 = 6 \), confirming our solution. Validation is an excellent habit to form in mathematics as it builds confidence in problem-solving.

Steps for Solving Equations

Solving equations is a methodical process that involves several steps, ensuring a structured approach to finding the solution. Let's break it down:

• Step 1: Isolate the Variable: You want to work towards getting the unknowns on one side of the equation. For radical equations like one with fourth roots, isolate the radical term first as it makes your next steps simpler.
• Step 2: Eliminate the Radical: Once you have the radical isolated, it's time to remove it. For fourth roots, raise both sides to the power of 4. This removes the radical sign and simplifies the equation.
• Step 3: Solve for the Variable: With the radical term removed, solve the remaining linear equation by isolating \( x \).
• Step 4: Verify the Solution: Substituting the found value back into the original equation ensures that your solution is correct. It is the final vital step in solving equations.

By following these steps in sequence, you’ll have a clear, logical method for arriving at the correct solution. It also makes it easier to identify where you might have gone wrong if you don’t end up with a verified solution.