Solving \$\sqrt[5]{3x-8}=3\$ A Step-by-Step Guide

Hey guys! Let's dive into how to solve this radical equation step by step. We'll break it down so it's super easy to understand. Our mission? To find the real number solution for the equation ${ \sqrt[5]{3x-8} = 3 }$

So, grab your thinking caps, and let's get started!

Understanding the Equation

Before we jump into solving, let's understand what we're dealing with. We have a fifth root here, which means we're looking for a number that, when raised to the power of 5, will give us the expression inside the root. The equation is:

${ \sqrt[5]{3x-8} = 3 }$

The left side involves a fifth root, and our goal is to isolate ${ x }$. To do that, we need to undo the fifth root. How do we do that? By raising both sides of the equation to the power of 5. This is a crucial step because it helps us eliminate the radical and simplify the equation.

Why Raise to the Power of 5?

Raising to the power of 5 is the inverse operation of taking the fifth root. Think of it like this: if you have a square root, you square it to get rid of the root. Similarly, for a fifth root, you raise it to the fifth power. This allows us to get rid of the radical sign and work with a simpler equation.

Potential Pitfalls

One thing to keep in mind with radical equations is the potential for extraneous solutions. These are solutions that pop up during the solving process but don't actually satisfy the original equation. This is more common with even roots (like square roots) because they can introduce sign changes. However, since we're dealing with an odd root (a fifth root), we don't have to worry about extraneous solutions in this case. Odd roots behave nicely and don't introduce these tricky situations.

Now that we've got a handle on the equation and the concept behind it, let's move on to the next step: actually solving it!

Solving the Equation Step-by-Step

Alright, let's get down to the nitty-gritty and solve this equation. We're starting with:

${ \sqrt[5]{3x-8} = 3 }$

Step 1: Eliminate the Fifth Root

To get rid of the fifth root, we need to raise both sides of the equation to the power of 5. This is the key step in unraveling the equation:

${ (\sqrt[5]{3x-8})^5 = 3^5 }$

When we raise the left side to the power of 5, the fifth root and the power of 5 cancel each other out. This leaves us with:

${ 3x - 8 = 3^5 }$

On the right side, we need to calculate ${ 3^5 }$. This means 3 multiplied by itself five times:

${ 3^5 = 3 \times 3 \times 3 \times 3 \times 3 = 243 }$

So our equation now looks like this:

${ 3x - 8 = 243 }$

Step 2: Isolate the Variable Term

Our next goal is to isolate the term with ${ x }$. To do this, we need to get rid of the -8 on the left side. We can do this by adding 8 to both sides of the equation:

${ 3x - 8 + 8 = 243 + 8 }$

This simplifies to:

${ 3x = 251 }$

Step 3: Solve for ${ x }$

Now we're in the home stretch! To solve for ${ x }$, we need to get it all by itself. Since ${ x }$ is being multiplied by 3, we'll divide both sides of the equation by 3:

${ \frac{3x}{3} = \frac{251}{3} }$

This gives us:

${ x = \frac{251}{3} }$

And there you have it! We've found our solution. Now, let's make sure this is a real number solution and that it makes sense in the context of our original equation.

Verifying the Solution

Okay, so we've arrived at a solution: ${ x = \frac{251}{3} }$. But before we pump our fists in victory, it's crucial to verify that this solution actually works. Plugging our solution back into the original equation ensures that we haven't made any mistakes along the way and that our solution isn't an extraneous one. Remember, even though we're dealing with an odd root and extraneous solutions are less of a concern, it's still a good practice to verify.

Our original equation is:

${ \sqrt[5]{3x-8} = 3 }$

Let's substitute ${ x = \frac{251}{3} }$ into the equation:

${ \sqrt[5]{3(\frac{251}{3})-8} = 3 }$

Step 1: Simplify Inside the Root

First, we simplify the expression inside the fifth root. Notice that the 3 in the numerator and the 3 in the denominator cancel out:

${ \sqrt[5]{251-8} = 3 }$

Now, subtract 8 from 251:

${ \sqrt[5]{243} = 3 }$

Step 2: Evaluate the Fifth Root

Next, we need to find the fifth root of 243. We're asking ourselves,