Problem 8
Question
Solve for \(x\) and check. $$2=\sqrt[4]{1+3 x}$$
Step-by-Step Solution
Verified Answer
The solution to the equation is \(x = 5\), and this solution has been verified as correct.
1Step 1: Isolate the Radical Expression
Start by isolating the radical expression on one side of the equation. In the given equation, the radical expression, \(\sqrt[4]{1+3x}\), is already isolated on the right side.
2Step 2: Remove the Fourth Root
Raise both sides of the equation to the power of 4 to remove the fourth root. This gives us \(2^4 = (\sqrt[4]{1+3x})^4\), which simplifies to \(16 = 1 + 3x\).
3Step 3: Solve for x
Subtract 1 from both sides to isolate the term with \(x\) on one side. This results in \(16 - 1 = 1 + 3x - 1\), which simplifies to \(15 = 3x\). Now, divide both sides by 3 to solve for \(x\), giving \(x = 15 / 3 = 5\).
4Step 4: Check the Solution
Substitute \(x = 5\) back into the original equation to check if it satisfies the equation: \(2 = \sqrt[4]{1+3(5)} = \sqrt[4]{16}\). Since \(\sqrt[4]{16} = 2\), the solution \(x = 5\) is correct.
Key Concepts
Radical Expression IsolationExponentiation to Eliminate RadicalsChecking Mathematical Solutions
Radical Expression Isolation
Understanding how to isolate radical expressions is essential for solving equations that involve roots. In the case of our equation \[2=\sqrt[4]{1+3 x}\], the radical expression is already isolated on one side, which is the ideal starting point for the solution process.
When a radical expression is not isolated, you would typically perform algebraic operations such as adding, subtracting, multiplying, or dividing to move other terms to the opposite side of the equation. This step is crucial because it simplifies the equation and paves the way for the next phase, which is to eliminate the radical by using exponentiation.
When a radical expression is not isolated, you would typically perform algebraic operations such as adding, subtracting, multiplying, or dividing to move other terms to the opposite side of the equation. This step is crucial because it simplifies the equation and paves the way for the next phase, which is to eliminate the radical by using exponentiation.
Exponentiation to Eliminate Radicals
Exponentiation is the process of raising a number or expression to a power. It is used in solving radical equations to remove the radical sign. The key is to raise both sides of the equation to the power that corresponds to the degree of the root to ensure that both sides remain equal. In our example, we have a fourth root, so we raise both sides of the equation to the fourth power.
This step is represented mathematically as \[2^4 = (\sqrt[4]{1+3x})^4\], which simplifies to \[16 = 1 + 3x\]. The radical is eliminated, and we are left with an algebraic equation easier to solve. This technique is powerful but must be used carefully, as it can sometimes introduce extraneous solutions that must be checked in the original equation.
This step is represented mathematically as \[2^4 = (\sqrt[4]{1+3x})^4\], which simplifies to \[16 = 1 + 3x\]. The radical is eliminated, and we are left with an algebraic equation easier to solve. This technique is powerful but must be used carefully, as it can sometimes introduce extraneous solutions that must be checked in the original equation.
Checking Mathematical Solutions
Once we have found a potential solution to a radical equation, we must ensure it is a valid solution by substituting it back into the original equation. This is a crucial step, as sometimes the steps taken to solve an equation can introduce solutions that don't actually satisfy the original equation, known as extraneous solutions.
In our exercise, we check our solution by substituting \[x = 5\] back into the original equation: \[2 = \sqrt[4]{1+3(5)} = \sqrt[4]{16}\]. Since this checks out, as \[\sqrt[4]{16} = 2\], we can be confident that \[x = 5\] is indeed a valid solution. This step is an important habit to form, as it can prevent the acceptance of false solutions and ensure accuracy in mathematical problem-solving.
In our exercise, we check our solution by substituting \[x = 5\] back into the original equation: \[2 = \sqrt[4]{1+3(5)} = \sqrt[4]{16}\]. Since this checks out, as \[\sqrt[4]{16} = 2\], we can be confident that \[x = 5\] is indeed a valid solution. This step is an important habit to form, as it can prevent the acceptance of false solutions and ensure accuracy in mathematical problem-solving.
Other exercises in this chapter
Problem 7
Express in radical form. $$\left(\frac{x}{y}\right)^{-1 / 3}$$
View solution Problem 7
Simplify, and write without negative exponents. Do some by calculator. $$a^{3} b^{-2}$$
View solution Problem 8
Express in radical form. $$a^{0} b^{-3 / 4}$$
View solution Problem 8
Addition and Subtraction of Radicals. Combine as indicated and simplify. $$3 \sqrt[3]{108}+2 \sqrt[3]{32}-\sqrt[3]{256}$$
View solution