Problem 19
Question
Solve each equation by hand. Do not use a calculator. $$\sqrt[4]{x-2}+4=6$$
Step-by-Step Solution
Verified Answer
The solution to the equation is \( x = 18 \).
1Step 1: Isolate the Radicals
Start by isolating the fourth root term in the equation. Subtract 4 from both sides to get: \( \sqrt[4]{x-2} = 2 \).
2Step 2: Eliminate the Fourth Root
To eliminate the fourth root, raise both sides of the equation to the fourth power: \( \left( \sqrt[4]{x-2} \right)^4 = 2^4 \).
3Step 3: Simplify the Equation
Simplifying both sides gives: \( x - 2 = 16 \).
4Step 4: Solve for x
Add 2 to both sides to solve for \( x \): \( x = 18 \).
Key Concepts
Fourth RootIsolating RadicalsSolving Equations by Hand
Fourth Root
The concept of a fourth root is similar to that of a square root, but instead of finding a number that when multiplied by itself twice equals the original number, we are looking for a number that when multiplied by itself four times equals the original number. It is denoted as \( \sqrt[4]{a} \), which can also be expressed as \( a^{1/4} \).
Understanding how to work with the fourth root is crucial, especially when solving equations that involve radical expressions, such as \( \sqrt[4]{x-2} \).
When you have a radical equation with a fourth root, the key to solving it is to eliminate the radical first. This is typically done by raising the entire equation to the fourth power. This process is vital for isolating the term under the radical so that you can continue solving for the unknown variable.
Understanding how to work with the fourth root is crucial, especially when solving equations that involve radical expressions, such as \( \sqrt[4]{x-2} \).
When you have a radical equation with a fourth root, the key to solving it is to eliminate the radical first. This is typically done by raising the entire equation to the fourth power. This process is vital for isolating the term under the radical so that you can continue solving for the unknown variable.
Isolating Radicals
Isolating radicals is often the first step when solving radical equations. It means rearranging the equation so that the radical expression stands alone on one side of the equation. This allows us to focus directly on removing the radical.
Consider the equation \( \sqrt[4]{x-2}+4=6 \). The initial goal here is to isolate the fourth root. To do this, simply subtract 4 from both sides, resulting in \( \sqrt[4]{x-2} = 2 \). Having the radicals isolated makes it easier to perform further operations.
Getting the radical term by itself helps in determining what math operations to perform next, such as raising both sides to the corresponding power to eliminate the root.
Consider the equation \( \sqrt[4]{x-2}+4=6 \). The initial goal here is to isolate the fourth root. To do this, simply subtract 4 from both sides, resulting in \( \sqrt[4]{x-2} = 2 \). Having the radicals isolated makes it easier to perform further operations.
Getting the radical term by itself helps in determining what math operations to perform next, such as raising both sides to the corresponding power to eliminate the root.
Solving Equations by Hand
Solving equations by hand is a valuable skill especially for understanding the underlying math concepts without relying on a calculator. To solve by hand involves breaking down each operation step by step until the unknown variable is isolated.
After isolating the radical, \( \sqrt[4]{x-2} = 2 \), the next step is to eliminate the fourth root by raising both sides of the equation to the fourth power. This process removes the root and results in \( (2)^4 \).
Upon simplifying, this becomes \( x - 2 = 16 \). The final step involves simple arithmetic: add 2 to both sides to solve for \( x \). Thus, \( x = 18 \). These sequential steps require attention to detail and reinforce the foundational math skills needed for solving radical equations manually.
After isolating the radical, \( \sqrt[4]{x-2} = 2 \), the next step is to eliminate the fourth root by raising both sides of the equation to the fourth power. This process removes the root and results in \( (2)^4 \).
Upon simplifying, this becomes \( x - 2 = 16 \). The final step involves simple arithmetic: add 2 to both sides to solve for \( x \). Thus, \( x = 18 \). These sequential steps require attention to detail and reinforce the foundational math skills needed for solving radical equations manually.
Other exercises in this chapter
Problem 19
Find all complex solutions for each equation by hand. $$1+\frac{3}{x}=\frac{5}{x^{2}}$$
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For function find all asymptotes and the coordinates of any holes in its graph. $$f(x)=\frac{x^{2}-2 x+3}{x+3}$$
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Evaluate each expression. $$\left(-9^{3 / 4}\right)^{2}$$
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Explain how the graph of \(f\) can be obtained from the graph of \(y=\frac{1}{x}\) or \(y=\frac{1}{x^{2}} .\) Draw a sketch of the graph of \(f\) by hand. Then
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