Problem 43 - Elementary and Intermediate Algebra

Question

Solve each equation. Let \(f(x)=\sqrt[4]{3 x+1}\). For what value(s) of \(x\) is \(f(x)=4 ?\)

Step-by-Step Solution

Answer

The solution is \(x = 85\).

Step 1: Set up the equation

We start by setting up the equation given in the problem: \(f(x) = \sqrt[4]{3x + 1} = 4\). We need to solve for \(x\).

Step 2: Remove the fourth root

To eliminate the fourth root, raise both sides of the equation to the fourth power: \((\sqrt[4]{3x + 1})^4 = 4^4\), which simplifies to \(3x + 1 = 256\).

Step 3: Solve for x

Subtract 1 from both sides to isolate the term with \(x\): \(3x = 256 - 1\), which simplifies to \(3x = 255\).

Step 4: Divide to find x

Divide both sides by 3 to solve for \(x\): \(x = \frac{255}{3}\), which simplifies to \(x = 85\).

Key Concepts

Fourth Roots

When we talk about a fourth root, we refer to a mathematical operation in which you want to determine what number, when raised to the fourth power, would give you a certain value. Mathematically, the fourth root of a number \( a \) is the number \( b \) such that \( b^4 = a \). This is similar to finding a square root, but instead of dividing the exponent into pairs, you divide it into fours. Fourth roots are a form of radical expression. They provide a way to reverse the process of raising a number to the power of 4. Just like square roots allow us to find the base \( b \) when we know \( b^2 \), fourth roots help us find the base when we know \( b^4 \).

• Common notation for fourth root is \( \sqrt[4]{a} \).
• Finding the fourth root can be crucial in solving equations that contain power terms.

Algebraic Functions

An algebraic function is a function that involves only algebraic operations such as addition, subtraction, multiplication, division, and taking roots. In our exercise, the function \( f(x) = \sqrt[4]{3x + 1} \) is an algebraic function, since it involves taking the fourth root of a linear expression.Algebraic functions can often be manipulated and solved through basic algebraic procedures:

• The function's form allows us to apply operations that help with finding unknown variables.
• These functions are central to a wide variety of mathematical solutions.

In solving equations with algebraic functions, it is crucial to understand how to manipulate the expressions to isolate unknown variables. This often involves operations such as factoring, distributing, and working with radicals.

Isolation of Variables

Isolation of variables is a fundamental concept in solving equations. It involves rearranging an equation so that the unknown variable is by itself on one side of the equation. This tends to make finding the value of the variable much simpler.In our specific example of solving the equation for \( x \), we began with \( (3x + 1) \), a part of the expression within the fourth root. By eliminating the root and systematically using algebraic operations, we were able to isolate \( x \):

• Subtract 1 from both sides to simplify the equation.
• Divide both sides by the coefficient of \( x \) to completely isolate \( x \).

This process is universally applicable to many types of equations, not just those involving fourth roots. The key to isolation of variables is maintaining the equation's balance by applying the same operations on both sides.