Problem 43 Solve equation. \(\sqrt[4]{x+1... [FREE SOLUTION]

Chapter 8: Problem 43

Solve equation. \(\sqrt[4]{x+12}=\sqrt[4]{3 x-4}\)

Short Answer

Expert verified

The solution to the equation is x = 8.

Step by step solution

Eliminate the Fourth Root

Raise both sides of the equation to the fourth power to eliminate the roots. This gives \( (\root{4}{x+12})^4 = (\root{4}{3x-4})^4 \). Simplifying both sides, you get \( x + 12 = 3x - 4 \).

Rearrange the Equation

Rearrange the equation to isolate the variable x on one side. Subtract x from both sides: \( 12 = 2x - 4 \).

Solve for x

Add 4 to both sides to further isolate the term with x: \( 16 = 2x \). Then, divide both sides by 2 to solve for x: \( x = 8 \).

Verify the Solution

Substitute your solution back into the original equation to verify: \( \root{4}{8 + 12} = \root{4}{3(8) - 4} \) simplifies to \( \root{4}{20} = \root{4}{20} \), which is true.

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Key Concepts

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

Fourth Roots

A fourth root of a number is a value that, when raised to the power of four, equals the original number. For example, the fourth root of 16 is 2 because \(2^4 = 16\). In algebra, finding fourth roots often involves working with radical expressions, which can be rewritten as exponential expressions. When solving equations involving fourth roots, we sometimes need to eliminate the radicals by raising both sides of the equation to the fourth power, thereby simplifying the equation.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to isolate the desired variable. In the given problem, we start by eliminating the fourth roots by raising both sides of the equation to the fourth power: \[ (\root{4}{x+12})^4 = (\root{4}{3x-4})^4 \] which simplifies to \[ x + 12 = 3x - 4 \].
At this stage, we perform algebraic operations to isolate the variable x. First, we subtract x from both sides, resulting in \[ 12 = 2x - 4 \].
Next, we add 4 to both sides: \[ 16 = 2x \].
Finally, we divide by 2 to solve for x: \[ x = 8 \]. By following these steps, we systematically solve the equation.

Verification of Solutions

Verification is a crucial step in solving algebraic equations. It ensures that our solution satisfies the original equation. For the equation \(\root{4}{x+12} = \root{4}{3x-4} \), we substitute \ x = 8 \ back into the original equation. This gives us: \ \root{4}{8 + 12} = \root{4}{3(8) - 4} \.
Simplifying both sides yields: \ \root{4}{20} = \root{4}{20} \.
Since both sides are equal, our solution \ x = 8 \ is verified as correct. Verification helps prevent errors and confirms that our steps were accurately followed.

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91影视

Solve equation. \(\sqrt[4]{x+12}=\sqrt[4]{3 x-4}\)

Short Answer

Step by step solution

Eliminate the Fourth Root

Rearrange the Equation

Solve for x

Verify the Solution

Key Concepts

Fourth Roots

Algebraic Manipulation

Verification of Solutions

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