91Ó°ÊÓ

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.

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.

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.