Problem 29 - Intermediate Algebra

Question

Solve. \(\sqrt[4]{4 x+1}-2=0\)

Step-by-Step Solution

Answer

The solution is \( x = \frac{15}{4} \).

Step 1: Isolate the Radical Expression

First, we need to isolate the fourth root on one side of the equation. The given equation is \( \sqrt[4]{4x+1} - 2 = 0 \). We can start by adding 2 to both sides of the equation to get \( \sqrt[4]{4x+1} = 2 \).

Step 2: Eliminate the Radical

To eliminate the fourth root, raise both sides of the equation to the power of 4: \( (\sqrt[4]{4x+1})^4 = 2^4 \). This simplifies to \( 4x + 1 = 16 \).

Step 3: Solve for x

Now, solve for \( x \) by first subtracting 1 from both sides: \( 4x = 15 \). Then divide both sides by 4 to get \( x = \frac{15}{4} \).

Key Concepts

fourth root

The term "fourth root" refers to a specific type of radical expression. A fourth root asks, "What number multiplied by itself four times will equal this number?" For example, the fourth root of 81 is 3, because \(3 \times 3 \times 3 \times 3 = 81\).
In our equation, \(\sqrt[4]{4x+1}\), the fourth root seeks a value that, when raised to the fourth power, results in \(4x+1\).
Understanding these roots is crucial when solving equations because we often need to manipulate them to simplify and ultimately solve for the unknown variable. This manipulation is done by removing the radical, which leads us to isolating and eliminating the radical expression.

isolate the radical expression

Isolating the radical expression is a key step in solving radical equations. In general, it involves moving all other terms to the opposite side of the equation, leaving the radical expression by itself.
In our exercise, we begin with the equation \(\sqrt[4]{4x+1} - 2 = 0\). To isolate the radical, we add 2 to both sides, resulting in \(\sqrt[4]{4x+1} = 2\).
This step is essential because isolating the radical simplifies the equation, making it easier to manage later steps, such as raising both sides to a power to remove the radical.

eliminate the radical

Eliminating the radical is an important subsequent step after isolating it. The goal is to remove the uncertainty or complexity created by the radical.
To eliminate the fourth root in the equation \(\sqrt[4]{4x+1} = 2\), we raise both sides to the fourth power. This step leverages the property of radicals and exponents, where a fourth root can be neutralized by raising it to the fourth power.
After this operation, our equation becomes \((\sqrt[4]{4x+1})^4 = 2^4\), simplifying to \(4x + 1 = 16\).
This process of elimination transforms the equation into a simpler, linear form that can easily be solved for the variable \(x\).

step-by-step solution

A step-by-step solution is invaluable for understanding how to tackle complicated equations. Let's break it down:

• Start with isolating the radical by manipulating the equation. This typically involves basic operations like addition or subtraction.
• Next, eliminate the radical by applying an appropriate exponent. For fourth roots, use the fourth power.
• After eliminating the radical, solve the resulting equation for the unknown variable.

In our example, we accomplish these steps by isolating the term, \(\sqrt[4]{4x+1} = 2\), then eliminating the fourth root by raising both sides to the fourth power. This yields \(4x + 1 = 16\).
Finally, solve for \(x\) by performing simple algebra: subtract 1, then divide by 4, resulting in \(x = \frac{15}{4}\). Each step systematically simplifies the problem, making complex operations approachable and manageable.