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Chapter 10: Problem 29

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

Short Answer

Expert verified
x = \frac{80}{3}

Step by step solution

01

Isolate the radical

First, isolate the term containing the fourth root. To do this, add 4 to both sides of the equation:\[\sqrt[4]{3x+1}-4+4=-1+4\]This simplifies to:\[\sqrt[4]{3x+1}=3\]
02

Eliminate the fourth root

Raise both sides of the equation to the power of 4 to eliminate the fourth root:\[(\sqrt[4]{3x+1})^4=3^4\]This simplifies to:\[3x + 1 = 81\]
03

Solve for x

Subtract 1 from both sides to isolate terms involving x:\[3x + 1 - 1 = 81 - 1\]This simplifies to:\[3x = 80\]Finally, divide both sides by 3:\[x = \frac{80}{3}\]

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Key Concepts

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

Isolate the Radical
Before we can solve the equation, we need to isolate the term that contains the root. Isolating the radical simplifies our work. In the given equation, \[\root[4]{3x+1} - 4 = -1\], the term \[\root[4]{3x+1}\] is the radical expression.
To isolate it:
  • Add 4 to both sides
This gives:
\[\root[4]{3x+1} - 4 + 4 = -1 + 4\]
Which simplifies to:
\[\root[4]{3x+1} = 3\]
Now, the radical is isolated, making the next steps easier.
Eliminate the Root
Once the radical is isolated, it's time to eliminate the root by raising each side of the equation to the appropriate power.
In our example, we need to eliminate a fourth root, so we raise both sides to the power of 4:
\[\root[4]{3x+1}^4 = 3^4\]
This cancels out the root on the left-hand side:
\[\root[4]{3x+1} \rightarrow (3x+1)^1 = 81\]
Simplifying gives us a new equation without any radicals:
\[3x + 1 = 81\]
Solve for x
With the radicals removed, we move forward by solving for x.
First, subtract 1 from both sides to isolate terms involving x:
\[3x + 1 - 1 = 81 - 1\]
Which simplifies to:
\[3x = 80\]
Finally, to solve for x:
  • Divide both sides by 3:
\[x = \frac{80}{3}\] Now, we have our solution, \[x = \frac{80}{3}\]. By following these steps, we have successfully solved the radical equation.

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