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Chapter 10: Problem 30

Solve. $$\sqrt[4]{2 x+3}-5=-2$$

Short Answer

Expert verified
x = 39

Step by step solution

01

Isolate the Radical

First, we need to isolate the radical. Start by adding 5 to both sides of the equation to get rid of the -5 on the left side: ewline ewline \ \ \ \(\sqrt[4]{2x + 3} - 5 + 5 = -2 + 5\) ewline ewline This simplifies to: ewline ewline\( \ \ \sqrt[4]{2x + 3} = 3 \)
02

Eliminate the Radical

Next, eliminate the fourth root by raising both sides of the equation to the fourth power: ewline ewline \( \ \ \ \ \left( \sqrt[4]{2x + 3} \right)^4 = 3^4 \) ewline ewline This simplifies to: ewline ewline \( \ \ \2x + 3 = 81 \)
03

Solve for x

Now, solve for x by isolating it. Start by subtracting 3 from both sides: ewline ewline \(\ \ 2x + 3 - 3 = 81 - 3 \) ewline ewline This simplifies to: ewline ewline\(\ \ 2x = 78 \) ewline ewline Then, divide both sides by 2: ewline ewline \( \ \ \ \ \ \ \frac{2x}{2} = \frac{78}{2} \) ewline ewline Which gives: ewline ewline\( \ \ \x = 39 \)

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Key Concepts

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

Isolating the Radical
When solving radical equations, the first step is always crucial: isolating the radical on one side of the equation. In our example, we start with the equation \(\sqrt[4]{2x+3}-5=-2 \).

To isolate the radical, we need to remove any additional terms on the same side as the radical. Here, the term \(-5 \) is on the left side with the radical.

Add 5 to both sides of the equation: \[ \sqrt[4]{2x+3} - 5 + 5 = -2 + 5 \]

This simplifies to: \[ \sqrt[4]{2x+3} = 3 \]

Successfully isolating the radical means you can now focus on eliminating it, allowing for easier manipulation of the terms.
Raising to a Power
At this stage, you have the equation with the radical isolated: \(\sqrt[4]{2x + 3} = 3 \).

To remove the fourth root, you need to raise both sides of the equation to the fourth power. This step is known as 'raising to a power.'

When you do that, the equation transforms as follows: \[ \left( \sqrt[4]{2x + 3} \right)^4 = 3^4 \]

The left side simplifies by eliminating the radical, resulting in: \[ 2x + 3 = 81 \]

Raising to a power is an efficient way to eliminate radicals and allows you to work with a straightforward polynomial equation.
Solving for a Variable
After eliminating the radical, you are left to solve for the variable in the simplified equation: \(2x + 3 = 81 \).

Follow these steps:
  • First, isolate the term containing the variable by subtracting 3 from both sides:
    • \[ 2x + 3 - 3 = 81 - 3 \]

    This simplifies to: \[ 2x = 78 \]
  • Next, solve for x by dividing both sides by 2:
    • \[ \frac{2x}{2} = \frac{78}{2} \]

    This simplifies to: \[ x = 39 \]

    Breaking the problem down into these smaller steps ensures that solving for the variable becomes a more manageable task.
Understanding and applying these steps methodically guarantees that you can solve radical equations with accuracy.

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