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Chapter 7: Problem 32

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

Short Answer

Expert verified
x = 39.

Step by step solution

01

Isolate the Radical Expression

Start by isolating the radical expression on one side of the equation. For the given equation \ \( \sqrt[4]{2x + 3} - 5 = -2 \), add 5 to both sides to isolate the radical. \ \( \sqrt[4]{2x + 3} = 3 \)
02

Eliminate the Radical

Raise both sides of the equation to the power of 4 to eliminate the fourth root. \ \[ (\sqrt[4]{2x + 3})^4 = 3^4 \ 2x + 3 = 81 \ \]
03

Solve for x

Subtract 3 from both sides to get the term involving \ x \ alone. \ \( 2x = 78 \). Then, divide both sides by 2 to solve for \ x \. \ \[ x = 39 \ \]
04

Check the Solution

Substitute \ x = 39 \ back into the original equation to ensure it is correct. \ \[ \sqrt[4]{2(39) + 3} - 5 = -2 \ \sqrt[4]{81} - 5 = -2 \ 3 - 5 = -2 \ -2 = -2 \ \] Since both sides are equal, \ x = 39 \ is indeed correct.

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Key Concepts

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

Isolating the Radical
To solve radical equations, we need to get the radical expression alone on one side of the equation. This is the first and crucial step. For the given problem, \( \sqrt[4]{2x + 3} - 5 = -2 \), we start by isolating the fourth root. We add 5 to both sides: \( \sqrt[4]{2x + 3} = 3 \). Now, the radical \( \sqrt[4]{2x + 3} \) is by itself. This makes it easier to deal with.
Eliminating the Radical
After isolating the radical, the next step is to eliminate it. To do this, we raise both sides of the equation to the power that matches the radical. Since it's a fourth root, we raise both sides to the power of 4: \[ (\sqrt[4]{2x + 3})^4 = 3^4 \] This simplifies to \[ 2x + 3 = 81 \] since raising a fourth root to the power of 4 cancels out the radical. This step converts the radical expression into a simple linear equation, making it easier to solve for x.
Checking Solutions
After finding the value of x, we must check our solution to ensure it is correct. This step verifies that the solution satisfies the original equation. Substitute \( x = 39 \) back into the original equation: \[\root[4]{2(39) + 3} - 5 = -2 \] Simplify inside the root: \[\root[4]{81} - 5 = -2 \] Since \( \root[4]{81} = 3 \), we have \[\root[4]{81} - 5 = 3 - 5 \] which equals \[-2 \]. Since both sides of the equation are equal, \( x = 39 \) is indeed the correct solution. This final check confirms the accuracy of our earlier steps.

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