Problem 46 - Precalculus : Building Concepts and Connections

Question

Solve the radical equation to find all real solutions. Check your solutions. $$\sqrt[4]{2 x+1}=3$$

Step-by-Step Solution

Answer

The solution to the equation \(\sqrt[4]{2 x+1}=3\) is \(x = 40\).

Step 1: Remove the fourth root

To remove the fourth root, we raise both sides of the equation to the power of four. This gives us the equation \( (2x+1) = 3^4 = 81\).

Step 2: Solve for the unknown

Now we solve for \(x\). We first subtract 1 from both sides of the equation to get \(2x = 81 - 1 = 80\). Then we divide both sides by 2 to isolate \(x\), giving us the equation \(x = 80 / 2 = 40\).

Step 3: Check the solution

We substitute \(x = 40\) back into the original equation to check the solution: If \( \sqrt[4]{2*40+1} = \sqrt[4]{81} = 3\) holds true, then the solution is correct.

Key Concepts

Understanding the Fourth Root

Grasping the concept of the fourth root is crucial when working with radical equations. The fourth root of a number is a value that, when raised to the power of four, gives the original number as the result. It is denoted as \( \sqrt[4]{x} \) and is the opposite operation of raising a number to the fourth power.

For example, in the equation \( \sqrt[4]{2x+1}=3 \) we are looking for what number, when multiplied by itself four times, would produce \(2x+1 \) such that the equation equals 3. To eliminate the fourth root and solve for \( x \) we raise both sides to the power of four. This step utilizes the principle that a root and a power operation with the same index cancel each other out, simplifying the equation and allowing us to move closer to finding the value of \( x \).

Isolating the Variable

When solving equations, the goal is often to 'isolate the variable,' which means to get the variable on one side of the equation and everything else on the other. To isolate \( x \) in our equation, we perform inverse operations, which are operations that reverse the effect of previous steps. In our case, after removing the fourth root by raising both sides of the original equation to the power of four, we proceed by performing operations aimed at leaving \( x \) on its own.

Let's walk through the steps:

Subtract 1

After removing the fourth root, we get \( 2x+1 = 81 \). The next step involves subtracting 1 from both sides: \( 2x = 80 \).

Divide by 2

Next, to completely isolate \( x \) we divide both sides by 2, yielding \( x = 40 \). Now \( x \) is isolated, and we have a proposed solution for the radical equation.

Checking the Solution

It's vital to verify any solution derived from an equation involving radicals. The process of raising both sides to a power when solving radical equations can sometimes introduce extraneous solutions, which are results that don't satisfy the original equation. To check, we substitute our proposed solution back into the original equation.

In our example, we insert \( x = 40 \) back into \( \sqrt[4]{2x+1} \) to see if we achieve the same value on both sides of the equal sign. We calculate \( \sqrt[4]{2*40+1} = \sqrt[4]{81} \), and since \( \sqrt[4]{81} = 3 \) we confirm that \( 3 = 3 \) which validates our solution. By checking, we ensure that our answer is indeed a real solution to the original radical equation.