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Chapter 1: Problem 62

Solve each equation. $$\sqrt[4]{3 x+1}=1$$

Short Answer

Expert verified
The solution to the equation is \( \textcolor{blue}{x = 0} \).

Step by step solution

01

Understand the problem

The equation given is \(\textcolor{blue}{\frac{1}{4}} \sqrt[4]{3 x + 1} = 1 \). This means we need to find the value of \(\textcolor{blue}{x} \) that makes this equation true.
02

Eliminate the fourth root

To eliminate the fourth root, we raise both sides of the equation to the power of 4. That is \((\textcolor{blue}{\frac{1}{4}} \sqrt[4]{3 x + 1})^4 = (1)^4\). This simplifies to \(\textcolor{blue}{3 x + 1} = 1\).
03

Solve for x

Now, solve for \(\textcolor{blue}{x}\) by isolating the variable. Subtract 1 from both sides: \(\textcolor{blue}{3 x + 1 - 1} = 1 - 1\), which simplifies to \(\textcolor{blue}{3 x} = 0\). Finally, divide by 3: \(\textcolor{blue}{x} = 0 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fourth Root
The fourth root of a number is what you multiply by itself four times to get that number. For example, the fourth root of 16 is 2 because \(2 \times 2 \times 2 \times 2 = 16\). It鈥檚 written as \(\text{鈭� } ^{4} x\) or \(x^{1/4}\) in math. In our exercise, we have \(\frac{1}{4} \sqrt[4]{3 x + 1} = 1\). To solve this, we first address the fourth root. By raising both sides to the power of 4, the fourth root is removed because \( (\text{ \sqrt } ^{4} (3x+1))^{4} = 3x + 1 \). This leaves us with the simplified equation to solve. Raising to the fourth power cancels out the fourth root, which is key in solving the equation. This simplification is important in precalculus problems.
Isolating Variables
In solving equations, a common step is isolating the variable. This means getting the variable alone on one side of the equation, making it easy to see its value. In our example, after removing the fourth root, we鈥檙e left with the equation \(3 x + 1 = 1\). To isolate \(x\), we need to move other numbers away from \(x\). First, subtract 1 from both sides, simplifying to \(3 x = 0\). Then, divide both sides by 3 to find that \(\textcolor{blue}{x} = 0\). Isolating the variable is an essential step in solving linear equations and many other types of equations in precalculus.
Precalculus
Precalculus serves as the foundation for calculus, covering topics and concepts that will be essential later. It combines algebra, geometry, and trigonometry to prepare students. From quadratic equations to logarithms and root functions, precalculus covers a broad range of topics. In this exercise, we dealt with root functions and linear equations, both common in precalculus. Understanding these foundational concepts in precalculus makes learning calculus and other advanced math topics easier later on. Practice and familiarity with concepts like isolating variables and manipulating roots are crucial for success.

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